A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A....

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ORIGINAL ARTICLE

A tribute to the use of minimalisticspatially-implicit models of savanna vegetation

dynamics to address broad spatial scales inspite of scarce data

Ivric Valaire Yatat Djeumenab Alexis Tchuinte Tamenclowast Yves Dumontade Pierre CouteroncfaUniversity of Pretoria Department of Mathematics and Applied Mathematics Pretoria South Africa

bUniversity of Yaounde 1 National Advanced School of Engineering Yaounde CamerooncIRD Umr AMAP Lmi DYCOFAC Yaounde Cameroon

alexistchuinteyahoofr yatatvalairegmailcom ivricyatatdjeumenupaczadCIRAD Umr AMAP Pretoria South Africa

eAMAP Universite de Montpellier CIRAD CNRS INRA IRD Montpellier Franceyvesdumontciradfr yvesdumontupacza

fAMAP IRD Cirad CNRS INRA Universite de Montpellier Montpellier Francepierrecouteronirdfr

lowastThe first two authors contributed equally

Received 5 September 2018 accepted 16 December 2018 published 20 December 2018

AbstractmdashThe savanna biome encompasses a va-riety of vegetation physiognomies that traduce com-plex dynamical responses of plants to the rain-fall gradients leading from tropical forests to hotdeserts Such responses are shaped by interactionsbetween woody and grassy plants that can be eitherdirect disturbance-mediated or both There hasbeen increasing evidence that several vegetationphysiognomies sometimes highly contrasted maydurably coexist under similar rainfall conditionssuggesting multi-stability or at least not abrupt tran-sitions These fascinating questions have triggeredburgeoning modelling efforts which have howevernot yet delivered an integrated picture liable to

furnish sensible predictions of potential vegetationat broad scales In this paper we will recall thekey ecological processes and resulting vegetationdynamics that models should take into account Wewill also present the main modelling options presentin the literature and advocate the use of minimalisticmodels capturing only the essential processes whileretaining sufficient mathematical tractability andrestricting themselves to a minimal set of parametersassessable from the overall literature

Keywords-Biogeography Rainfall Fires Ordi-nary differential equations Impulsive differentialequation Tree-Grass interactions Multi-stability

Copyright copy 2018 Yatat Djeumen et al This article is distributed under the terms of the Creative Commons AttributionLicense (CC BY 40) which permits unrestricted use distribution and reproduction in any medium provided the originalauthor and source are creditedCitation Ivric Valaire Yatat Djeumen Alexis Tchuinte Tamen Yves Dumont Pierre Couteron A tributeto the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatialscales in spite of scarce data Biomath 7 (2018) 1812167 httpdxdoiorg1011145jbiomath201812167 Page 1 of 29

IV Yatat Djeumen A Tchuinte Tamen Y Dumont P Couteron A tribute to the use of minimalistic

I INTRODUCTION

Savannas have been identified by biogeogra-phers as a biome corresponding to warm meanannual temperatures (gt 20degC) and a broad rangeof intermediate mean annual rainfall (100ndash2000mmyrminus1) (Sarmiento (1984) [76] Youta Happi(1998) [114] Abbadie et al (2006) [1] Lehmannet al (2011) [58]) Such climatic context predom-inates along rainfall gradients leading from sub-equatorial wet climates to hot arid climates Thewider definition to which we refer here tends tointegrate climatic variant allowing for nearly desertvegetation or on the contrary seasonal tropicalforests Savannas display specific interplays ofnatural constraints that prevent or at least impedeclosure of woody cover and ensuing suppressionof light-demanding herbs and grasses A centralalbeit non-exclusive cause for this is the rsquoideal fireclimatersquo (Trollope (2011) [98]) that characterizestropical regions with seasonal droughts alternatingwith warm and wet rainy seasons producing highherbaceous biomass that once dried-up becomeshighly ignitable and fuels fires (Frost et al (1986)[39] Thonicke et al (2001) [93] Govender et al(2006) [44]) High frequency of lightning stormswhich is a characteristic of Africa (Abbadie et al(2006) [1] Trollope (2011) [98]) also contributesto make it the rdquoFire continentrdquo even though presentfire regimes mostly rely on human-made ignitions(Archibald et al (2009) [7] Govender et al (2006)[44] Trollope (2011) [98])

Dynamics of vegetation within the savannabiome has long interested ecologists as it clearlydeparts from the classical post-disturbance succes-sion pathways that are expected to rapidly bringback closed canopy forest as observed in most oftemperate and wet tropical climatic zones (Bond etal (2005) [22]) The last decades have witnessedburgeoning efforts of modelling as to account forthe possibly long-lasting coexistence of grassy andwoody components and try to predict potentialshifts from two-phased vegetation physiognomiesThese efforts have however not yet delivered anintegrated picture liable to furnish at broad scales(ie for fractions of continents) sensible predic-

tions of possible vegetation dynamics Such a bigpicture is nevertheless desirable for figuring outthe future of vegetation in the face of climate andanthropic change scenarios (Mayaux et al (2004)[60] Bond et al (2005) [22] Archibald et al(2009) [7] Accatino et al (2010) [4] Favier et al(2012) [36]) It is also necessary for applicationsto territories devoid of reference data and long-term observation sites as it is the case for most oftropical Africa

The objectives of the present contribution arefourfold It first aims at recalling and synthetizingthe main array of facts about ecological processesand resulting vegetation dynamics that modelsshould aim to capture and render (see section II)Second in order to claim genericity we synthetizethe main modelling options present in the liter-ature and put emphasis on minimalistic modelscapturing only essential processes while retainingsufficient mathematical tractability and restrictingthemselves to a minimal set of assessable pa-rameters (see section III) Thirdly on this basiswe argue that such models have now becomemore comprehensive and useful for meaningfulpredictions (see section IV) Finally we discusshow those models may now help guiding datacollection for improved calibration and testing ofdynamical hypotheses (see section V)

II A BRIEF REVIEW ON SPACE-IMPLICIT

TREE-GRASS INTERACTIONS MODELLING

A Tree-grass coexistence and possible alternativestable states

Over very large tropical territories field ob-servers have documented long lasting coexistenceof notable levels of grass and woody biomass(Backeus (1992) [9]) The most frequently re-ported form of coexistence is observed locallythrough vegetation physiognomies that associatefairly continuous grassy cover and more or lessscattered populations of trees and shrubs of vary-ing clumping levels This is referred to as sa-vanna physiognomy (see Figure 1) Such vege-tation types mixing both lifeforms are manifoldand progressively merge in space or through time

Biomath 7 (2018) 1812167 httpdxdoiorg1011145jbiomath201812167 Page 2 of 29

without clear-cut boundaries (Torello-Raventos etal (2013) [95]) Another modality of long lastingassociation between herbaceous and woody life-forms occur at landscape scale under the form ofmosaics featuring forests (usually closed canopyones) and open savannas or grasslands (eg Figure1 Bond and Parr (2010) [21]) In those landscapesthat pertain to moist-wet climates normally seenas favourable to forests the mosaics appear highlycontrasted and among the most rdquoemblematic vege-tation transitionsrdquo in the world (Oliveras and Malhi(2016) [72]) outside the closed forest woodyvegetation is of low biomass and the dominantphysiognomies relate mainly to grassland More-over boundaries between forest and grassland aregenerally sharp (Hoffmann et al 2012 [48] Cuni-Sanchez et al (2016) [27])

Our interpretations of those various physiog-nomies are limited by the length of the observationwindows we can rely on for distinguishing trendsagainst fluctuations For field observations thiswindow length barely extend over some decadesand this only for a very small number of siteswhere invaluable data have been gathered At thescale of extensive territories representativeness ofthose sites remains yet an open question Remotesensing is progressively broadening our observa-tional means But the best nowadays space-bornesensors for estimating woody cover (Buccini andHanan (2007) [26]) or biomass (Mermoz et al(2014) [64] Bouvet et al (2018) [24]) are recentand do not allow tracking changes far back More-over the accuracy of those estimations notablyfor woody cover is limited due to the difficultyto separate grass vs tree in signal responses inmixed stands This is particularly true regard-ing long diachronic series that mainly featureoptical images of insufficient spatial resolutionApart from blatant changes eg forest encroach-ment or recession Mitchard and Flintrop (2013)[68] subtle evolution of the grass-tree balancein mixed physiognomies are still beyond reachRemote sensing however recently brought twointeresting contributions to the savanna debateFirst broad scale assessment of woody cover at

regional (Central Africa Favier et al (2012) [36])to continentalglobal scales (Hirota et al (2011)[47]) clearly showed that contrasted levels of covercan coexist under the same ranges of climaticconditions making the existence of multi-stablestates at least plausible Second in both Centraland West Africa comparison between ancient airphotographs from the 50s and satellite imagesfrom the 80-90s frequently evidenced a progress offorest over savannasgrasslands in landscape fea-turing contrasted mosaics of the type exemplifiedin Figure 1 (Youta Happi (1998) [114] Mitchardet al (2011) [69])

Even though there is still no conclusive evidencethat alternative stable states may exist within thesavanna biome models should be able to accountfor them as plausible outcomes of tree-grass in-teractions The same applies to savanna physiog-nomies locally associating trees and grasses thatmay be seen as either stable or transient two-phase states Since those mixed physiognomies areobservable at broad scale there is no reason to apriori rule out that some observed mixtures maybe stable under their local environmental contextIndeed hypothesis testing is a fundamental roleof models though this use is not so widespreadin ecology And to this aim the wider the arrayof reasonable predictions the more relevant is themodel

B Lines of thoughts

Most authors agree on the fact that soil waterbudget herbivory (ie grazing andor browsing)and fires are the principal factors influencinggrowth of woody and herbaceous plants and theirdynamical interactions (Scholes and Archer (1997)[79] Higgins et al (2000) [45] Scholes (2003)[78] Van Langevelde et al (2003) [101] Bondet al (2005) [22] Bond (2008) [17] Abbadie etal (2006) [1] Accatino et al (2010) [4] Staverand Levin (2012) [84] Baudena et al (2010) [11](2014) [12] Jeffery et al (2014) [51]) Authorshowever diverge on the relative importance ofthose factors in shaping dynamical outcomes oftree-grass interactions This is not surprising con-sidering the broad extent of the savanna biome

Figure 1 Landscape-scale mosaic between dense forest and herbaceous savanna (grassland) observed in central Cameroon (Ayos) Brown-

pink tree crowns indicate marshy forests in talwegs Note the weak congruence between topography and the occurrences of forest vs grassland

Airborne photo from N Barbier June 2017

and the variety of both environmental conditionsand anthropogenic pressures that apply therein Afactor appearing pervasive in a given context is notsystematically due to prevail elsewhere One groupof authors has been insisting on direct interactionsamong or between plant-types (ie tree-tree ortree-grass) such as competition for light or forsoil limiting resources (often moisture via rootsystems) (eg Scholes and Archer (1997) [79]Scholes (2003) [78]) It is obvious that the treendashgrass interaction is highly asymmetric trees havea strong competitive effect on grasses but grasseshave a weak competitive effect on mature treesalthough they may have a strong effect on saplingsthat have not grown above the grass layer (Scholes(2003) [78] Figure 2-a)

Another group of authors has been emphasizingthat woody vegetation would be likely to reach aclosed canopy situation and suppress grasses in theabsence of recurrent disturbances induced by firesor browsers (or both sources) that delay or blockthe build-up of woody biomass by destroying theaerial part of seedlings and saplings (eg Bondet al (2005) [22] Bond (2008) [17] Staver andLevin (2012) [84] Baudena et al (2010) [11](2014) [12] Jeffery et al (2014) [51] Figure 2-b amp -c) Browsers impact though undoubtedlypervasive in certain situations (McNaughton andGeorgiadis (1986) [62] Scholes and Walker (1993)

[80] Van Langevelde et al (2003) [101] Holdoet al (2009) [49]) is not systematic across thesavanna biome and the generality of the distur-bance hypothesis relies mainly on fire Indeedexperimental fire suppression systematically leadsto the thickening-up of the woody vegetation andto the development of dense woodlands or thicketsFor sufficient annual rainfall shifts toward closecanopy forests are also observed (Bond et al(2005) [22] Jeffery et al (2014) [51])

Literature may sometimes overemphasize thedistinction between rsquointeractionrsquo (between planttypes for limited resource) and rsquodisturbancersquo hy-potheses (see Scholes and Archer (1997) [79])as to make them appear as alternatives thoughthey are by no means mutually exclusive It iswidely acknowledged that to have notable impacton vegetation fire disturbance requests sufficientintensity through enough dry grass biomass asmain source of fuel Under a certain level ofgrass biomass owing to insufficient rainfall orintense grazing fires tend to spread difficultlyand where occurring have modest impacts onwoody plants Logically most authors tend nowto distinguish disturbance-limited (ie under moistclimate) vs water limited (ie arid) savannas (egBond et al (2003) [20]) Inter-tree competitionshapes the second type (Sankaran et al (2005)[75]) while asymmetric and fire-meditated tree-

Figure 2 Three facets of woody plant resprouting just after fire and rainfall onset in the humid savannas of the Sanagha basin (Cameroon

Central Africa) Note that tufts of perennial grasses did also systematically resprout Seedling struggling in a middle of a grass tuft a)

Seedling resprouting after topkill either at ground b) or stem c) level Photos Pierre Couteron (March 2018)

grass interactions is central to the first one Butless clear-cut situations obviously occur underintermediate rainfall (Diouf et al (2012) [31])or because of modulation by edaphic conditionsgrazing and anthropogenic pressures Grazing maylead savannas toward physiognomies and function-ing looking less fire-prone ie more rdquoarid-likerdquothan expected from the only climate features as anemergent consequence of dynamical amplificationof external forcing

III MAIN PUBLISHED MODELLING OPTIONS

The questions raised by observed or putativedynamics within the savanna biome have trig-gered an increasing interest in terms of modellingPioneering works (Walker et al (1981) [103]Walker and Noy-Meir (1982) [104]) first usedsystems of ordinary differential equations (ODE)to address the particular case of arid fire-immunesavannas in which excessive grazing fosters bushencroachment (Skarpe (1990) [82]) This line ofmodelling featured grass and woody biomasses asstate variables and aimed at explicitly depictingtheir interactions in relation to soil moisture dy-namics As such it became a paradigm for rsquointer-action modelsrsquo involving a limited resource but

the central assumption of soil niche partitioningbetween the two plant forms called Walterrsquos (1971)hypothesis [105] has been ever since hotly debatedand is obviously not verified in all ecologicalcontexts where savannas dry thickets or grasslandsare observable

Another line of ODE-based modelling built onthe application to savannas of the initial concept ofasymmetric competition of (Tilman (1994) [94])through a simple framework that allows consid-ering both direct and disturbance-mediated plantinteractions Tilmanrsquos framework reinterpretation(see Accatino et al (2010) [4]) De Michele etal (2011) [28] used two states variables namelycover-fractions of grass (G) and tree (T ) assumedexclusive and summing between zero and one Itmodelled their interacting dynamics in a system oftwo ODE

dT

dt= cTT (1minus T )minus δTT

dG

dt= cGG(1minus T minusG)minus cTTGminus δGG

(1)

where T and G are dimensionless and denotethe fractions of sites occupied by tree and grass

respectively cT and cG are the colonisation ratesof tree and grass respectively δT and δG representthe mortality rates of tree and grass respectivelyIn the sequel we refer to system (1) as Tilmanrsquosmodel

Logistic growth of the inferior competitor(grasses plus herbs) is bounded and depressedby the cover of the superior competitor (woodyplants) which logistic growth is not directly af-fected by grasses (asymmetric competition) Insystem (1) there is no fire-mediated retroactionof G on T This was however introduced bysubsequent authors (Van Langevelde et al (2003)[101] Beckage et al (2009) [14] Accatino et al(2010) [4] Beckage et al (2011) [13] De Micheleet al (2011) [28]) via a linear function of G Thusexplicitly including the impact of fire on T inTilmanrsquos model the first equation of (1) becomes

dT

dt= cTT (1minus T )minus δTT minus δF fω(G)T (2)

where δF represents the trees vulnerability to firef is the fire frequency (inversely proportional tofire return time period) and ω(G) is a functionof grass biomass that represents the fire impactThrough ω(G) there is thus indirect fire-mediatednegative feed-back of grass cover onto tree coverthat counterbalance direct tree-grass asymmetricinteractions

A larger array of models (see Tables I and II)took a leaf from the previous modelling frame-work (system 1 and equation 2) Main sourcesof variations between models were (1) natureof the equations and temporal treatment of firedisturbance (time-continuous forcing ie ODE vstime-discrete or impulsive occurrences) (2) natureof the function expressing grass-fire feedback ontrees (linear vs nonlinear) (3) integration of her-bivory in addition to fire (4) facultative explicittreatment of water availability through modelswith one (and sometimes more) additional statevariables expressing water resource in interactionwith vegetation variables We will refer to suchmodels as rsquoecohydrologicalrsquo (see Table I) amongwhich is system (3) proposed by Accatino et al

(2010) [4] that features a first equation devoted tothe dynamics of a soil moisture variable (S)

dS

dt=

p

w1(1minusS)minusεS(1minusTminusG)minusτTST

minusτGSG

dT

dt=cTST (1minusT )minusδTTminusδF fTω(G)

dG

dt=cGSG(1minusTminusG)minuscTSTGminusδGGminusfG

(3)

wherep

w1(per year) represents the rainfall rate

normalized with respect to root zone capacity ε(per year) is the evaporation τT and τG (per year)are water uptake parameters for tree and grassrespectively cT cG δT δG δF and f are definedas in system (1) and equation (2)

Note that setting ω(G) = 0 in the second equa-tion of (3) makes Tilmanrsquos model (1) analogousto the system coupling the second and the thirdequations of (3) Moreover if the S variable is heldconstant the main difference between systems (1)and (3) is that Accatino et al (2010) [4] consideredω(G) = G (ie impact of fire on trees as alinear function of grass biomass) while in Tilmanrsquosmodel this function is equal to zero (no impact offire)

Taking ω(G) as any increasing function of thegrass cover provides a more general expression ofthe fire impact on trees Without loss of generalitywe referred to Holling type functions (Holling(1959) [50] Augier et al (2010) [8] Tewa et al(2013) [92] see equation (4) for generic ones)The general form of ω(G) reads as

ω(G) =Gθ

Gθ + αθ (4)

where G in tons per hectare (thaminus1) is grassbiomass α is the value takes by G when fireintensity is half its maximum and the integer θdetermines the steepness of the sigmoid Non-linear response was retained by some other authors(Scheiter and Higgins (2007) [77] Higgins et al(2010) [46] Staver et al (2011) [83] Touboul et

al (2018) [96]) Accatino and De Michele (2016)[3] also introduced a piecewise linear functionof grass biomass (indeed qualitatively mimickingextreme non-linearity) in their non-equilibriummodel (NEM) as the probability of the occurrenceof fire

Considering a nonlinear (sigmoidal) shape forω(G) allows for the existence of up to three tree-grass coexistence (ie savanna) equilibria whiletwo of them may be simultaneously stable (iebistability) and forest-savanna-grassland tristabil-ity is reachable (Yatat et al (2014) [112] (2018)[109] Tchuinte Tamen et al (2014) [91] (2017b)[88]) Conversely we proved that for linear ω(G)functions tristability is unreachable (Yatat et al(2018) [109] Tchuinte Tamen et al (2018) [88])As pointed out in Yatat et al (2018) [109] possibletristability is in good agreement with results ofFavier et al (2012) [36] obtained along a generalclimatic transect over central African (latitude inthe range of 3 ndash 4 north) These results concerneda very large range of woody cover (wc) variations(from very low values approaching grassland phys-iognomies to nearly 80 cover (ie forest) throughwc of 40 ie savanna) which suggests grass-landsavannaforest tristability as at least plausibleODE models have been criticized on the basis thatthey only predict abrupt transitions between twoalternative stable states (Accatino and De Michele(2016) [3]) that are deemed unrealistic Howevertristability of equilibria as well as bistability oftwo savanna equilibria suggests that shifts fromone stable state to another may be less spectacularthan hypothesized from previous models and thatmodels may render more complex pathways ofvegetation changes (see Yatat et al (2018) [109]for further discussion)

In our earlier works (Yatat et al (2014) [112](2017) [110] (2018) [109] Tchuinte Tamen etal (2014) [91] (2016) [89] (2017) [90]) wechose to use above-ground biomasses instead ofcovers as state variables in contrast to Accatinoet al (2010) [4] and most of the models whichhave been proposed on the subject (reinterpretationof Tilman (1994) [94] Baudena et al (2010)

[11] Staver et al(2011) [83] De Michele et al(2011) [28] Synodinos et al (2015) [85]) thatconsidered cover fractions Modelling biomasseshelp accounting from the fact that plant typesare not mutually exclusive at a given point inspace since field studies suggested that grass oftendevelop under tree crowns (Belsky et al (1989)[16] Belsky (1994) [15] Weltzin and Coughenour(1990) [107] Abbadie et al (2006) [1] Dohn etal (2012) [33] Moustakas et al (2013) [70])Moreover biomasses directly refer to the cycleof carbon and can be assessed from radar remotesensing in savanna ecosystems that correspond towoody biomasses below the saturating level of thebackscatter L-band radar signal (Mermoz et al(2015) [65] Bouvet et al (2018) [24])

The minimal configuration of the publishedmodels (Tables I amp II) featured only two vege-tation state variables (eg Van Langevelde et al(2003) [101] Tchuinte Tamen et al (2014) [91](2016) [89] (2017) [90] (2017b) [88] Synodinoset al (2018) [86]) as in Tilmanrsquos (1994) [94] initialframework but several models distinguished sizeclasses within the woody component of vegetation(eg Favier et al (2004) [37] Baudena et al(2010) [11] Staver et al (2011) [83] Yatat etal (2014) [112] (2017) [110] Touboul et al(2018) [96]) Some models used more than twosize classes through matrix population models (egAccatino et al (2013) [2] (2016) [5] and refer-ences therein) Simpler models used only two size-related variables in addition to grass and simplyaccount for the asymmetric nature of tree-grassinteractions as discussed previously (see also Sc-holes (2003) [78] Yatat et al (2014) [112] (2017)[110]) Other models separate large trees havingtop buds above the flame zone and therefore facinglimited risks of topkill from smaller trees andshrubs which have high probability of having theiraerial systems destroyed (Beckage et al (2009)[14] Staver et al (2011) [83] Yatat et al (2014)[112] (2017) [110]) This distinction stems fromfield observations (Trollope (1984) [97] Trollopeand Trollope (1996) [99]) that evidence rapiddecline of percent topkill with tree height (see

Table I Comparison of several models of treendashgrass dynamics with respect to some modelling optionsWalterrsquos hypothesis refers to the differences in root depth of herbaceous and woody vegetation in waterseeking while ecohydrological frameworks stand for models that consider additional state variablesexpressing water resource in interaction with vegetation variables From Tchuinte Tamen (2017) [87]and Yatat (2018) [113] The symbol lowast means that we refer to system (1)

State Tree Herbivory Walterrsquos Ecohydro-

Authors variables state variables perturbation hypothesis logical

Cover Biomass All sizes lumped Size-structured applied frameworks

Walker et al (1981) [103] X X X X

Tilman (1994)lowast [94] X X

Higgins et al (2000) [45] X X X

Van Langevelde et al (2003)[101] X X X X

DrsquoOdorico et al (2006) [32] X X

Beckage et al (2009) [14] X X

Baudena et al (2010) [11] X X

Accatino et al (2010) [4] X X X

De Michele et al (2011) [28] X X X X

Staver et al (2011) [83] X X

Yu and DrsquoOdorico (2014) [115] X X X X

Touboul et al (2018) [96] X X

Synodinos et al (2018) [86] X X X X

Yatat et al (2014) [112] X X X

Tchuinte Tamen et al (2014)[91] X X X

Tchuinte Tamen et al (2018)[88] X X X X

also Figure 3) ODE models featuring two woodyvariables in addition to grass proved analyticallytractable (Beckage et al (2009) [14] Staver etal (2011) [83] Yatat et al (2014) [112] (2017)[110]) as long as other complexities were notintroduced

A strong objection against ODE models is that

fire is not a forcing that continuously removes asmall fraction of biomass through time as per theprevious ODE equation systems Instead fire ac-tually suppresses a substantial fraction of biomassat once through punctual outbreaks that shapeecosystem aspect and immediate post-fire func-tioning (Figure 4) This principle was implemented

Table II Summary of the characteristics of treendashgrass interactions models with respect to fire modellingoptions (continued Table I) From Tchuinte Tamen (2017) [87] and Yatat (2018) [113] Some references(unticked) do not model fire The symbol lowast means that we refer to system (1)

Fire perturbation Impact of Fire

Authors time- time- time- time- linear sigmoidalcontinuous stochastic discrete impulsive forms

Walker et al (1981) [103]

Tilman (1994)lowast [94]

Higgins et al (2000) [45] X X

Van Langevelde et al (2003) [101] X X

DrsquoOdorico et al (2006) [32] X

Accatino et al (2010) [4] X X

De Michele et al (2011) [28] X X

Yu and DrsquoOdoricco (2014) [115] X X

Touboul et al (2018) [96] X X X

Synodinos et al (2018) [86] X X

Yatat et al (2014) [112] X X

Tchuinte Tamen et al (2014) [91] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

Tchuinte Tamen et al (2018) [88] X X X

via time-discrete recurrence equation models byScheiter and Higgins (2007) [77] Higgins (2010)[46] But another framework of impulsive differ-ential equations (IDE) also proved relevant to gainrealism regarding nature and consequences of firewhile keeping a high level of analytical tractabilitythanks to the ODE modelling of inter-fires vegeta-tion dynamics (Yatat et al (2017) [110] TchuinteTamen et al (2016) [89] (2017) [90])

IV REACHING SENSIBLE PREDICTIONS FROM

MINIMAL MODELS

A A seminal rdquobig picturerdquo at biogeographic scale

The model from Accatino et al (2010) [4]was pioneering and inspiring in that it first ven-tures into generically predicting vegetation phys-iognomies (in terms of percent covers of woody vsherbaceous plants) over the entire savanna biomeThe authors used bifurcation analysis (Accatino

0

10

20

30

40

50

60

70

80

90

100

HEIGHT CLASSES minus m

TO

PK

ILL

minus

0 minus 050 051 minus 100 101 minus 150 151 minus 200 201 minus 250 251 minus 300 301 minus 350 351 minus 400 401 minus 450 451 minus 500

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

Figure 3 Illustration of the effect of height on the frequency of topkill of individual trees subjected to fires in the Kruger National Park

(panel (a)) and in the central highlands of Kenya (panel (b)) In the panel (b) continuous bars denote head fire while black-dash bars represent

back fire (reproduced from Trollope and Trollope (2010) [100] copy Trollope and Trollope (2010) [100] )

Figure 4 Aspects of two nearby savannas both located close to a forest boundary in the Sanagha Basin Cameroon (Central Africa)

depending on recent fire occurrence (left) or not (right) Photos were taken the same day On the left note the general resprouting of both

the herbaceous stratum and topkilled woody plants Photos Pierre Couteron March 2018

et al (2010) [4] Tchuinte Tamen et al (2018)[88] see also Figure 7) based on two importantparameters of strong intuitive meaning namelymean total annual rainfall and fire frequency Theythereby achieved delineation of domains in whichmain physiognomies (ie grassland savanna for-est) can be expected as stable Bistability situations(forest-grassland and forest-savanna) were alsohighlighted for sufficiently high fire frequenciesFor low fire frequencies a sensible gradient ofincreasing woody cover with increasing annualprecipitation was found But one may note thatonly dense woodlands (ie still two-phase vegeta-tion) were obtained even for the highest rainfallrange while forest stricto -sensu (mono-phasewith no tall light-demanding savanna grasses inthe understory) is widely observed for the cor-responding ranges of precipitations Moreovertransitions between vegetation types in relationto fire frequencies proved tricky Indeed in thehigh rainfall range increasing fire frequency leadsfrom the aforementioned woodlands to forestIn the intermediate rainfall range increasing firefrequency leads from savanna mono-stability toforest-savanna bistability (see Figure 7) Analo-gously for fairly low rainfall grassland stabilityshifts to grassland-forest bistability Hence allover the rainfall gradient it looks as if sufficientfrequency of fire were a necessary condition toreach forest (bi)-stability This is contradicted byempirical knowledge according to which frequentfires are known to jeopardize or at least delaywoody biomass build-up but never favour it (Bondet al (2005) [22] Archibald et al (2009) [7] Bondand Parr (2010) [21])

Where did this critical problem come fromMost of subsequent papers barely evoked the ques-tion A large share of them investigated differentmodelling options often more complex andorless tractable or they assumed particular biogeo-graphic conditions In a further contribution Ac-catino and De Michele (2016) [3] argued about in-trinsic limitations of ODE-based modelling Theyalso put forward that there is no evidence accord-ing to which observed vegetation physiognomies

may be close to a stable equilibrium point It isin fact undisputable that climate is likely to varythrough time and there is no guaranty that woodyvegetation can track such variation with enoughcelerity They also underline as questionable theassumption according to which the parameter fof fire frequency should be treated as a constantforcing independently of vegetation characteris-tics All these arguments brought them to proposea rsquonon-equilibrium modelrsquo based on stochasticdifference equations as alternative to the time-continuous model of Accatino et al (2010) [4]referred to as rsquoequilibrium-modelrsquo (EM) In theirnon-equilibrium (NEM) model fire occurrence isa stochastic event all the more likely to occur in agiven dry season that ignitable dry grass biomassabundantly built-up in the foregone rainy seasons

Accatino and De Michele (2016) [3] comparedthe predictions of their (EM) vs (NEM) modelsThey argued that separation of fire-immune vsfire-prone savannas as an indirect consequence ofrainfall is an emergent property with their NEMwhile it is artificially induced by the choice of thef parameter with EM They also pointed out thatwhen considering high rainfall the NEM is able toreproduce the rdquobimodalityrdquo of woody cover extentobserved in remote sensing studies But in facttheir NEM was not a straightforward time-discreteanalogue of the ODE based EM of Accatino etal (2010) [4] since it features several noveltiesTherefore the differences they reported betweenEM and NEM are not a simple consequence oftime-continuous fire forcing vs time-discrete fireoccurrences In fact several aspects altogethercontribute to the more satisfactory results obtainedwith the NEM by Accatino and De Michele(2016) [3] We will subsequently illustrate the factthat predictions that are qualitatively satisfactorycan be obtained by directly improving the ODEbased rsquoEMrsquo framework notably regarding the fire-mediated feedback of grass onto tree dynamics

B Fire frequency grass biomass and fire impact

The way in which fire impact is modelled in theprevious equation systems (1 and 3) is obviouslya crucial question As stated by Scholes (2003)

0 2 4 6 8 100

2

4

6

8

10

Grass biomass (tha minus1)

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

Figure 5 Graphical representation of the function shapes w(G) The fire impact term in the ODEs is given by λfT fω(G)

Table III Functions involving ω(G)

Functions Models (ODE) Coexistence

equilibria

ω(G) = 0 Tilman (1994) [94]One savanna

monostability

ω(G) = GAccatino et al (2010) [4] 2 savannas

Van Langevelde et al (2003) [101] bistability

Hollingtype II Tchuinte Tamen et al (2014) [91]

2 savannas

bistability

Hollingtype III

Staver et al (2011) [83] 3 savannas

Tchuinte Tamen et al (2014) [91] bistability

Yatat et al (2014) [112] 3 savannasbistability amptristabilityTchuinte Tamen et al (2018) [88]

[78] modelling savanna dynamics in fire-pronecontexts actually requires introducing an rdquoequationpredicting the effect of grass biomass via fireintensity on tree biomassrdquo

In fact non-linearity in ω(G) may be justifiedon various non-exclusive grounds since what isimportant is to properly model as a whole thecausal chain that leads from grass abundance andignition regime to woody biomass suppressionAs steps in this chain we may identify (i) firefrequency to be seen as an external forcing upon

the tree-grass system (think about a targeted fireregime in a managed area such as a ranch or aprotected area) (ii) actual yearly fire probability(or frequency) of occurrence in any arbitrary smallpiece of land once (i) has been set (iii) firepotential impact (intensity and flame height) onwoody vegetation (iv) fire actual impact that alsodepends on features intrinsic to woody vegetation(see below section IV-C)

Fire intensity which is strictly speaking a quan-tity of energy released (Bond and Keeley (2005)

Figure 6 Example of spatial heterogeneity of fire propagation in an altitude mosaic of forests and low biomass grasslands in Cameroon

Here local community hunters tend to set fire every year at landscape scale (ie f = 1) as to flush small game from spots of dense grass

cover But all the area does not burn every year because fire actually do not propagate everywhere (From P Couteron February 2017 Mount

Cameroon National Park)

[18]) appears empirically as a fairly linear functionof grass biomass But impact on trees also dependson flame height which is reported as increasingexponentially with observed quantity of dry grassbiomass (Scheiter and Higgins (2007) [77] Staveret al (2011) [83] Synodinos et al (2018) [86])though one may suppose some levelling off formaximal grass biomass (and height) values Inearlier works (Yatat et al (2014) [112] (2017)[110] (2018) [109] Tchuinte Tamen et al (2014)[91] (2016)[89] (2017) [90]) we systematicallyassumed fire impact on woody vegetation as anon-linear increasing bounded function of grassbiomass (Yatat et al (2014) [112] (2017) [110](2018) [109] Tchuinte Tamen et al (2014) [91](2016) [89] (2017) [90] (2018) [88]) w(G)

In our modelling the f parameter was kept asconstant multiplier of w(G) but we interpret itas a man-induced rdquotargetedrdquo fire frequency (as forinstance in a fire management plan) which will nottranslate into actual frequency of fires of notableimpact as long as grass biomass is not of sufficientquantity With this interpretation the actual fireregime may substantially differ from the targetedone as frequently observed in the field Andwhatever f values any hypothetical piece of land

will actually be fire-prone only if other forcing fac-tors (climate herbivory etc) allow for sufficientgrass biomass Most previous modelling papersincluding those from our group did not elaboratemuch regarding the successive steps involved inthe grass-fire feedback Distinction between (i)and (ii) may appear subtle and to our knowl-edge was never emphasized before It directly re-sults from space-implicit savanna modelling Fireregime which is nowadays overwhelmingly man-induced (Govender et al (2006) [44] Archibald(2009) [7]) is a forcing at landscape scale sincepeople do not go and set fire in every piece ofland They instead count on fire propagation thatdepends on abundance and spatial evenness of drygrass In presence of low and unevenly distributedgrassy fuel fire will barely propagate leaving alarge share of the area unburnt This makes thedifference between steps (i) and (ii) as frequentlyobserved in the field (see Diouf et al (2012) [31]Figure 6) The response of percent area burnt tograss abundance is likely to be sharply nonlinearas suggested by the impressive results reported byMcNaughton (1992) [61] at the scale of the entireSerengeti complex in Tanzania In this remarkablestudy local fire frequency dwindled over a decade

following grass biomass suppression by soaringherbivore populations while the ignition regime bypeople dwelling around the park likely remainedmore or less the same In fact since we are heredealing with mean-field models the ω(G) functionis also due to embody the difficult spreading of firein presence of fuel of overall low quantity keepingin mind natural spatial variability of grass biomass(Figure 6) Non-linearity of ω(G) seems thereforea necessary feature for adequately capturing thefire-mediated grass-tree feedback

C Tree survival

To be relevant the most parsimonious modelsfeaturing just grass and tree state variables mustovercome the limitation pointed out in sub-sectionIV-A for the precursor model of Accatino et al(2010) [4] All things being equal any increasein fire frequency should never increase the woodycomponent of vegetation Fire if any is expectedto be of no substantial consequence over the drieststretch of the rainfall gradient while for the moisterpart it is widely observed that extending theaverage time between successive fires (decreas-ing frequency) favours the building up of woodyvegetation Accounting for that proved to be achallenge for minimal two-variable models thatdo not distinguish between fire sensitive and fireinsensitive woody fractions Non-linearity of theω(G) function though important proved not suf-ficient to overcome this problem Tchuinte Tamenet al (2017) [90] further introduced a secondnon-linear decreasing function which directly ex-presses that high woody biomasses correspondingto tall trees proportionately experience far less fire-related losses than low woody biomasses relatingto seedlings saplings and shrubs (see Figure 2)We hence proposed the following function to de-note the fire-induced treeshrub mortality

ϑ(T ) = λminfT + (λmaxfT minus λminfT )eminuspT (5)

where λminfT (in yrminus1) is minimal loss of treebiomass due to fire in systems with a very largetree biomass while λmaxfT (in yrminus1) is maximal lossof treeshrub biomass due to fire in open vegetation

(eg for an isolated small woody individual havingits crown within the flame zone) p (in tminus1ha) isproportional to the inverse of biomass suffering anintermediate level of mortality

This general form was suggested by experimen-tal observations showing dwindling rate of topkillwith increasing tree height since tall trees arelikely to have their upper parts above the flamezone even for high grass biomass (Trollope andTrollope (2010) [100])

Notice that taking into account a nonlinear anddecreasing function of tree biomass is a way to by-pass introducing size classes as to keep the modelminimal and retain mathematical tractability (seeinspiring examples in Meron et al (2004) [67]Lefever et al (2009) [55]) The addition of theϑ(T ) function was indeed decisive in ensuring thata two-equation ODE system provides predictionsthat qualitatively agree with the general ecologicalknowledge about the role of fire return periodOn this basis Tchuinte Tamen et al (2017) [90]designed a model that also implemented punctualfire events through impulsive differential equa-tions But predictions of the ODE model itselfwere already satisfactory

D Relating to water resource

Several savanna models have explicitly mod-elled soil water resources via a dedicated equationas in system (3) But the soil moisture dynamicsis very rapid compared to change in vegetationSoil moisture variations linked to a given rainfallevent are damped within a few days (Barbier etal (2008) [10]) while vegetation growth proceedsover months for grasses and even years for woodyplants It therefore makes sense to consider veg-etation dynamics in relation to a level of soilwater resource that is approximately constant fora given level of total annual rainfall or ideallywater deficit (rainfall minus evapotranspiration)Martinez-Garcia et al (2014) [59] reached sim-ilar conclusions for a partial differential equa-tion model of non-local plant-plant interactionsfor water This justifies letting parameters in thevegetation dynamics equations directly depend on

climatic parameters This principle sustains themodel expressed through the following system

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

minusηTGTGminus λfGfG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

wherebull G and T are grass and tree biomasses respec-

tivelybull W is the mean annual precipitation (in mil-

limeters per year mmyrminus1)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bT +Ware annual productions of grass and treebiomasses respectively where γG and γT(in yrminus1) express maximal growths of grassand tree biomasses respectively half satura-tions bG and bT (in mmyrminus1) determine howquickly growth increase with water availabil-ity

bull KG(W ) =cG

1 + dGeminusaGW and KT (W ) =

cT1 + dT eminusaTW

are carrying capacities of

grass and tree respectively where cG andcT (in thaminus1) denote the maximum valuesof the grass and tree biomasses aG and aT(mmminus1yr) control the steepness of the curvesof KG and KT respectively and dG and dTcontrol the location of their inflection points

bull δG and δT respectively express the rates ofgrass and tree biomasses loss by herbivores(grazing andor browsing) or by human ac-tion

bull ηTG denotes the asymmetric influence of treeson grass for light (shading) and resources(water nutrients) which relate to competitiveor facilitative influences

bull λfG is the specific loss of grass biomass dueto fire

bull f = 1τ is the fire frequency where τ is the

fire return period

Submitting model (6) to bifurcation analysisprovides Figure 7-(b) that is to be compared toFigure 7-(a) from Accatino et al (2010) [4] whichhas been reobtained using Matcont (see Govaerts(2000) [42] Dhooge et al (2003) [29] Govaertset al (2007) [43] and references therein) BothFigures 7-(a) and 7-(b) are sensible regarding lowfire frequencies for which increasing MAR leadsto a sequence of physiognomies of increasingwoody biomass (ie grassland savanna forest)But in Figure 7-(b) the improvement resultingfrom introducing ω(G) and ϑ(T ) is apparent whenincreasing fire frequency (f ) at different levelsof the rainfall gradient For high MAR valuesthe expected physiognomy shifts from monostableforest to forest-grassland bistability Indeed inpresence of high MAR it is known that grasslandsare due to be encroached by forest under fireprevention or even just because of decreasingfire frequencies (Jeffery et al (2014) [51]) Ourmodel accords with field observations in that ahigh fire frequency is indeed a necessary conditionto perpetuate the grassland (or savannas of lowwoody biomass) physiognomies Moreover large-scale observations of bimodality between high andvery low woody cover situations (Hirota et al(2011) [47] Favier et al (2012) [36]) can beaccounted for by the forest-grassland bistabilitythough the converse is not necessarily true Infact bimodality may stem from either transientsituations or topographical heterogeneity and doesnot automatically implies bistability For low tointermediate MAR values say 600 minus 1000 mmfire is known to be less pervasive though fieldobservations or experiment results depict woodyvegetation thickening in case of fire frequencydecrease (Brookman-Amissah (1980) [25]) Themodel is able to render such thickening as ashift from the grassland to the savanna stabilitydomain The model also predicts forest-savanna orsavanna-grassland bistability and even tristabilitythereof for restricted domains in the MAR-firefrequency plane that were situated around 1000

mm of MAR This value is of course dependenton parameter values used for computations un-derlying figure 7-(b) Refined calibrations relatingto a specific regional context may displace thethresholds Notably the parameter expressing theinfluence of woody biomass on grasses proved tobe influential on the thresholds between vegetationstates (Tchuinte Tamen et al (2018) [88])

E Impulsive time-periodic occurrences of fireevents

In previous models the traditional time-continuous fire forcing formalism is often usedHowever it is questionable to model fire as apermanent forcing that continuously removes frac-tions of fire sensitive biomass all over the yearIndeed several months and even years can passbetween two successive fires such that fire maybe considered as an instantaneous perturbation ofthe savanna ecosystem (Yatat et al (2017) [110]Tchuinte Tamen (2016) [89] (2017) [90] see alsoTable IV page 18) Several recent papers haveproposed to model fires as stochastic events whilekeeping the continuous-time differential equationframework (Baudena et al (2010) [11] Beckageet al (2011) [13] Klimasara and Tyran-Kaminska(2018) [54] Synodinos et al (2018) [86]) or usinga time-discrete model (Higgins et al (2000) [45]Accatino and De Michele (2013) [2] Accatino etal (2016) [5]) However a drawback of most ofthe aforementioned recent time-discrete stochasticmodels (Higgins et al (2000) [45] Baudena et al(2010) [11] Beckage et al (2011) [13]) is that theybarely lend themselves to analytical approaches

Based on Table IV page 18 we further considerin our group (Yatat et al (2017) [110] (2018)[109] Yatat and Dumont (2018) [111] TchuinteTamen (2016) [89] (2017) [90] (2018) [88]) im-pulsive time-periodic fire events which is an ap-proximation that keeps the potential of analyticalinvestigation as large as possible while modellingdiscrete fires An impulsive differential equationssystem can be used to express fire through impul-

sive periodic occurrences (eg system (7) below)

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

∆G(nτ) = minusλfGG(nτ)

∆T (nτ) = minusϑ(T (nτ))ω(G(nτ))T (nτ)

n = 1 2 Nf

(7)

wherebull for π isin GT ∆π(nτ) = π(nτ+)minus π(nτ)

and π(nτ+) = limθrarr0+

π(nτ + θ)

bull τ = 1f is the period between two consecutive

firesbull Nf is a countable number of fire occurrencesbull nτ n = 1 2 Nf are called moments of

impulsive effects of fire and satisfy 0 le τ lt2τ lt lt Nfτ

Properties of models (6) and (7) have beenanalysed in Tchuinte Tamen et al (2018) [88]Below we provide some numerical simulations asto illustrate the bifurcations between the stabledomains delineated in Figure 7-b as a consequenceof increasing the fire frequency for two particularvalues of MAR ie W = 946 mmyminus1 and W =1003 mmyminus1 For each of the two MAR valueswe compare the consequences of increasing fwith the ODE (system (6)) and the IDE (system(7)) frameworks by comparing Figure 8 againstFigure 9 and Figure 10 against Figure 11 Forthe lower MAR situation ODE and IDE frame-works qualitatively agree and show the bifurcationfrom monostable savanna to monostable grasslandthrough an intermediate bistable situation (Figure8-b Figure 9-b) For the higher MAR case bothframeworks also show the transition from monos-table forest to forest-grassland bistability throughtristability involving savanna Qualitatively the

(a)

(b)

Figure 7 Bifurcation diagrams using Matcont (a) Accatino et al (2010) [4] model re-implementation (b) Implementation of system (6)

Single red green and black symbols (rectangles in panel (a) dots in panel (b)) stand for grassland forest and desert respectively Twinned red

and green symbols stand for savanna (coexistence state) Size of the symbols qualitatively denote grass and tree cover fractions in panel (a)

and biomass levels in panel (b) Parameters used to compute Figure 7-(a) are from Accatino et al (2010) [4] (see also Table V in appendix)

The parameter values used in 7-(b) are from Tchuinte Tamen et al (2018) [88] (see also Table VI in appendix) (Color in the online version)

Table IV Average fire period (τ in yr) ranges of values found in literature with respect of the meanannual rainfall (MAR) Ranges of values of MAR are from Yatat et al (2017) [110] and TchuinteTamen et al (2016) [89]

Ranges References

Low MAR 5 ndash 50 Frost and Robertson (1985) [40]

MAR le 650 mmyrminus1 4 ndash 8 Trollope (1984) [97]

Intermediate MAR 3 ndash 5 February et al (2013) [38]

650 mmyrminus1 le MAR le 1100 mmyrminus1 5 ndash 7 Van Wilgen et al (2004) [102]

High MAR 05 ndash 3 Jeffery et al (2014) [51]

MAR ge 1100 mmyrminus1 05 ndash 2 Bond and Keeley (2005) [18]

Accatino et al (2010) [4]

1 ndash 5 Abbadie et al (2006) [1]

1 Menaut and Cesar (1979) [63]

Gignoux et al (2009) [41]

predictions of the two frameworks agree about thepredicted sequence of vegetation physiognomieswhen increasing the fire frequency However theIDE model systematically predicted bifurcationsfor lower values of f than for the ODE Thisindicates that shifting to the conceptually moresatisfactory IDE framework will introduce speci-ficities in forthcoming stages concerning refinedcalibration and comparison with real-world obser-vations

V DISCUSSION AND PROSPECTS

In the present paper we emphasize that ecolo-gists did probably not yet exploit all the potentialof simple ODE systems for modelling vegetationdynamics in the savanna biome to which most sea-sonal tropical ecosystems pertain We showed thatreasonable non-trivial predictions can be obtainedin reference to hypothetical situations directly re-lating to rainfall and fire frequency gradients Ap-plication to specific contexts and locations wouldrequest refined calibration for the parameters ofthe generic minimalistic model But it may alsoinvite to better address specific processes deemedinfluential in a particular situation under studyThis would mean complexifying the model tomatch a specific piece of reality Though this is

actually a natural and sensible trend in scienceparsimony is an opposing principle that tells us tokeep complexification under control A meaning-ful balanced modelling approach should restrict towhat we strictly need to account for a well-definedarray of empirical facts in a particular situation

On the empirical side the ongoing developmentof remote-sensing techniques and derived productsis providing avenues to better depict the spatiotem-poral variation of environmental factors such asrainfall (eg via CHELSA Karger et al (2017)[52]) topography (via the SRTM Farr et al (2007)[35] published at increased spatial resolution byNASA in 2013) or fires (httpmodis-fireumdedueg Archibald et al (2009) [7] Diouf et al (2012)[31]) This also applies to the monitoring of somevegetation variables though disentangling effectson most remotely-sensed signals from grasses vstrees in mixed savanna physiognomies is stillchallenging Improving and diversifying sourcesof remote sensing information will obviously helpsorting out relevant predictions from unrealisticones and refine the benchmarking of modelsBut most of the parameters expressing vegetationdynamics will remain out of reach of remotely-sensed assessment and will remain dependent onfield information An increased effort of field data

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

Figure 8 Illustration of a bifurcation due to f with the ODE model of system (6) for a constant MAR value of W=946 mmyminus1 When

the fire frequency f increases the system shifts from a savanna monostable (see panel (a)) to a grassland monostable (see panel (c)) passing

through a bistability between savanna and grassland (see panel (b))

(a) (b) (c)

Figure 9 Illustration of a bifurcation due to fire frequency f with the impulsive model (7) for a constant MAR value of W=946 mmyminus1

This figure is based on same parameters values as Figure 8 but with the impulsive IDE framework savanna-grassland bistability is already

observable for f = 04 (panel (a)) and give way to monostable grassland for f=051 (panel (b)) These values are to be compared to f = 025

and f = 07 respectively when using the ODE model (system (6))

0 5 10 15 20Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

Figure 10 Numerical simulations with the ODE model (6) illustrating a bifurcation induced by increasing the fire frequency (f ) in

presence of a constant MAR (W) value of 1008 mmyminus1 Vegetation shifts from monostable forest (left) to savanna-forest bistability (center)

to grassland-savanna-forest tristability (right)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

Figure 11 Numerical simulations with the IDE model (7) illustrating a bifurcation induced by increasing the fire frequency (f ) in presence

of a constant MAR (W) value of 1008 mmminus1 This figure is analogous to Figure 10 obtained with the ODE model Note that with the ODE

model there is still tristability for f = 056 while forest is still monostable for f = 04

collection is obviously desirable but insufficientmeans for research in most tropical countries isenduring reality That strong data limitation is alasprobably here to stay finally pleads for parsimonyin modelling It also underlines the importance formodelling to be sufficiently convincing and acces-sible to ecologists as to guide data acquisition andorient scarce resources towards assessing param-eters proven as the most influential by sensibilityanalyses

On the modelling side within the class of mod-els that distinguished size classes for the woodycomponent of vegetation some used more thantwo size classes through matrix population models(eg Accatino et al (2013) [2] (2016) [5]) How-ever such models remain generally simulation-based and usually involve a fairly large number ofparameters Thus it is not easypossible to assesshow model parameter variations may influence themodel outcomes (Yatat et al (2018) [109]) ODE

and IDE models featuring two woody variablesin addition to grass proved analytically tractable(Beckage et al (2009) [14] Staver et al (2011)[83] Yatat et al (2014) [112] (2017) [110]) aslong as other complexities were not introducedFor example Yatat et al (2014) [112] (resp Yatatet al (2017) [110]) studied a ODE-like (respIDE-like) tree-grass interactions model where inaddition to grass they considered two classes ofwoody plants fire-sensitive like seedlings andfire insensitive But based on recent publicationsof our group we found that even with modelsthat feature only one state variable for woodycomponent meaningful results are obtained andto some extent are qualitatively similar to thoseobtained with models that used two size-relatedvariables for woody component (Tchuinte Tamenet al (2014) [91] (2016) [89] (2017) [90] (2018)[88] Yatat et al (2018) [109])

The IDE framework is an obvious improvementthat expresses a reasonable trade-off between in-creased realism and decreased analytical tractabil-ity Within the framework of IDE a further stepin that direction could be considering patterns offire occurrences featuring stochastic componentsinstead of the deterministic periodic regime weused (Yatat et al (2017) [110] (2018) [109]Tchuinte Tamen et al (2016) [89] (2017) [90](2018) [88]) But one may note here that peri-odicity of fire outbreaks is not that unrealisticin the context of subequatorial humid savannasfor which fires can only occur at the end of dryseasons which are of short duration Here theannual climatic cycle strongly defines the temporalwindow for fires while in less humid savannasfire is simultaneously less frequent and liable tooccur all over extensive dry seasons (Diouf et al(2012) [31]) We believe that while mathematicaltractability or theoretical study of a model is notan absolute requirement or is not always possibleit remains nevertheless desirable at least for tworeasons First it can appear as a kind of guaranteethat numerical simulations displayed by the modelare not the result of some numerical artifacts Inother words the choice or the construction of

a suitable algorithm to solve a given (complex)model strongly relies on its qualitative study orwhen this study is not possible on the analysisof some sub-models that can be mathematicallytractable Nowadays there are more and moreworks that point out some spurious behaviors thatmay appear when using some rsquoclassicalrsquo schemesfor model simulations (see for example Anguelovet al (2012) [6]) Second any theoretical analysismay provide useful informations about the role ofsome particular parameters in the dynamics of thesystem

We have here focused on spatially-implicit mod-els because we believe that such models havestill important insights to provide and also be-cause spatially-explicit models are far more de-manding in terms of parametrization and moredifficult to study theoretically Substantial effortsto design and run spatial models of savannashave however been made during the last decade(Borgogno et al (2009) [23]) Most of them re-lied on individual-based models such as cellularautomata At this step of the discussion it seemsmeaningful to point out that there are some authorswho pleaded for a mutualistic or complementaryrelationship between mathematical tractability-based and simulation-based formalisms (Omohun-dro (1984) [73] Wolfram (1985) [108] Weimar(1997) [106] Narbel (2006) [71] Dietrich et al(2014) [30] Dumont et al (2018) [34]) and wealso agree with that As an illustration it may bemore difficult to achieve a very deep theoreticalanalysis of a partial differential equations modelwhen taking into account spatial heterogeneitywhile it seems more easy to handle it when usingfor example an individual-based model or cellularautomaton formalism Independently partial dif-ferential equations (PDE) have also widely beenused to account for the genesis of vegetationregular spatial patterns (bare soil vs dense shrubbycover) in the particular context of arid savannas(Lefever and Lejeune (1997) [56] Klausmeier(1999) [53] Sherratt (2005) [81] Borgogno et al(2009) [23] Lefever et al (2009) [55] Meron(2011) [66] Lefever and Turner (2012) [57]) In

this line there was a recent attempt by Yatat et al(2018) [109] to account for the dynamics of forest-savanna boundaries by introducing local diffusionoperators for herbaceous and woody biomasses ina parsimonious space-implicit model with time-continuous fire events forcing closely related tothe one of Tchuinte Tamen et al (2017) [90]The analysis of this model shows that there ex-ists monostable or bistable travelling solutionsrelated to the boundary movements in the forest-grassland mosaic And the authors showed thatdepending on fire-return time as well as differencein diffusion coefficients of woody and herbaceousvegetation fire events are able to greatly slowdown or even stop the progression of forest in thehumid part of the savanna biome This kind ofresults obtained from theoretical analysis are ofgreat interest for practical needs or managementpolicies However as an improvement of Yatatet al (2018) [109] there are also some ongoingworks that aim to deal with existence of travellingwaves for system of Impulsive Partial DifferentialEquations (IPDE) that model savanna dynamics(eg Yatat and Dumont (2018) [111]) This type ofmodelling is of great interest considering that theforest-savanna ecotone is the most widespread inthe tropics and that forests have been encroachingduring the last decades in many humid savannasof West and Central Africa and to a lesser ex-tent in other regions of the world (Oliveras andMalhi (2016) [72]) On the other hand forestencroachment which is of great consequence forthe global carbon balance of terrestrial ecosystemsproved heterogeneous in space and time (Oliverasand Malhi (2016) [72]) Hence modelling shouldhelp better understand the hierarchy of processesand forces accounting for such heterogeneity Thismakes spatially-explicit modelling desirable Butinterpretation and calibration of local diffusionoperators as used in Yatat et al (2018) [109]cannot rely on much empirical knowledge in plantecology And one may note that this also appliesto colonization rates between adjacent cells thatare central to cellular automata models In thecase of PDEs modelling non-local plant-plant

interaction processes (eg through interaction ker-nels) is mathematically more challenging thoughattempts in Martinez-Garcia et al (2014) [59]and in Lefever et al (2009) [55] or Lefever andTurner (2012) [57] nevertheless provide sources ofinspiration

A recent line of criticism questions the rele-vance of reasoning in reference to equilibriumstates There is indeed no compelling evidence thatphysiognomies presently observable correspond oreven are close to predictable equilibria determinedby current environmental conditions In fact pa-rameters reflecting environmental variables no-tably climate are due to fluctuate or even changethrough time And vegetation especially its woodycomponent may be unable to keep pace withsuch variations and rather track them with delaythereby remaining distant from any equilibriumstate Long-lasting consequences of past climateperiods probably still mark present vegetationFor instance in Central Africa concomitant todrier period occurred some centuries ago (EuropersquosrdquoLittle Ice Agerdquo 500-200 years BP) which proba-bly provoked forest cover recession and fragmen-tation (Oslisly et al (2013) [74]) The trend ofwidespread forest boundaries displacement withinsavannas as observed during the last decades maybe a delayed recovery after this past drier periodthat is progressing at slow and unequal pace owingto the counteracting influence of fires IncreasedCO2 availability that favours C3 woody plantsagainst C4 savanna grasses (Bond and Midgley(2000) [19]) may also reinforce this trend

VI CONCLUSION

Minimal savanna models using ODE systemshave been criticized from different standpointsThe first one was the poor realism of the overallpicture made by the predicted stable equilibriaThe present paper shows that some unrealisticpredictions are not a direct drawback of theODE framework but rather derive from inadequatemodelling of the crucial fire-mediated negativefeedback of grassy biomass onto woody vegeta-tion Using nonlinear functions such as ω(G) andϑ(T ) (as in equations 5) is not only justified by the

nature of the mechanisms at play but also provedsufficient to get a meaningful rdquobig picturerdquo ofvegetation physiognomies predicted as stable forvarying mean annual rainfall and fire frequency (asin Figure 7-(b)) Another argument against ODEmodels is that they predict too contrasted stablestates meaning that shifts between them would ap-pear as more catastrophic than actually observedBut some strong contrasts such as landscape mo-saics of forest and grassland are indeed observablein the field (see Figure 1) Moreover equilibria thathave attraction domains rdquoadjacentrdquo in Figure 7 donot systematically show contrasted biomass valuesTransitions may actually be progressive in termsof state variables Indeed at low fire frequency thetransition from savanna to forest along the MARgradient corresponds to a continuous increase oftree biomass with concomitant decrease of grassbiomass The same applies to the transition fromsavanna to grassland via increased fire frequency(see Figures 8 and 9) Here the shape of the non-linear functions embodying the grass-fire feedbackmatters as shown in previous works (Yatat et al(2014) [112] Tchuinte Tamen et al (2014) [91])Strongly nonlinear shapes (eg Holling functionsof higher order see Table III) allow for multiplecoexistence equilibria (ie multiple savanna phys-iognomies) of different woody biomass valueswhich may be seen as rdquostepping stonesrdquo betweengrassland and forest Thus the ODE frameworkdoes not automatically imply abrupt changes be-tween equilibria of very contrasted biomass values

On the other hand it is undisputable that mod-elling fire as an external forcing continuouslysuppressing small amount of biomass through timeis not satisfactory Models based on punctual firesimpacting large shares of biomasses are morerelevant This is implemented in time-discretestochastic models which are however of limitedanalytical tractability Impulsive differential equa-tion systems are a good compromise since theypermit to model time-discrete fire impact whilekeeping ODE for modelling vegetation growth Assuch they remain analytically tractable to a largeextent while being more realistic

In this paper we show that minimal savannamodels are able to provide a wide array ofmeaningful and relevant predictions of savannadynamics while retaining sufficient mathematicaltractability and restricting themselves to a minimalset of parameters assessable from the overall litera-ture Moreover simplicity is overarching whateverthe level of tractability With a simple modelsimulations can claim a thorough exploring of allparameters space Conversely it is difficult to besure that sufficient exploration of model behaviorshas been carried out for overcomplicated modelswhich tend to flourish in ecology Moreover be-cause there is naturally substantial uncertainty formany parameter values it is difficult to concludewhether results are due to the ranges taken forparameters or to the structure of the model itselfTherefore using complex models it becomes evenmore illusory to test hypotheses while this is oneof the fundamental roles of modelling

ACKNOWLEDGEMENTS

A Tchuinte Tamen and IV Yatat Djeumenare grateful to the French National Institute forResearch for Sustainable Development (IRD) theFrench Agricultural Research Centre for Interna-tional Development (CIRAD) the French Institutefor Research in Computer Science and Automation(INRIA EPI MASAIE) the Department of Coop-eration and Cultural Action (SCAC) of the FrenchEmbassy in Yaounde the International Center forPure and Applied Mathematics (CIMPA) the In-ternational Laboratory for Computer Sciences andApplied Mathematics (LIRIMA) the annual inter-national conference on Mathematical Methods andModels in Biosciences (BIOMATH 2016 amp 2017)for their financial supports during the preparationof their PhD theses IV Yatat Djeumen alsoacknowledges the support of the SARChI Chairin Mathematical Models and Methods in Bioengi-neering and Biosciences (University of PretoriaSouth Africa)

REFERENCES

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[3] F Accatino and C De Michele Interpreting woodycover data in tropical and subtropical areas Compar-ison between the equilibrium and the non-equilibriumassumption Eco Comp 2560ndash67 2016 URLhttpdxdoiorg101016jecocom201512004

[4] F Accatino C De Michele R Vezzoli D Donzelliand R Scholes Tree-grass co-existence in savannainteractions of rain and fire J Theor Biol 267235ndash242 2010 URL httpdxdoiorg101016jjtbi201008012

[5] F Accatino K Wiegand D Ward and C De MicheleTrees grass and fire in humid savannas The impor-tance of life history traits and spatial processes EcolModell 320135ndash144 2016 URL httpdxdoiorg101016jecolmodel201509014

[6] R Anguelov Y Dumont and JM Lubuma Onnonstandard finite difference schemes in biosciencesAIP Conf Proc 1487212ndash223 2012

[7] S Archibald DP Roy B Van Wilgen and RJScholes What limits fire an examination of driversof burnt area in southern africa Global Change Biol15613ndash630 2009 URL httpdxdoiorg101111j1365-2486200801754x

[8] P Augier C Lett and JC Poggiale Modelisationmathematique en ecologie Cours et exercices corrigesDunod 2010

[9] I Backeus Distribution and vegetation dynamics ofhumid savannas in Africa and Asia J Veg Sci3(3)345ndash356 1992 URL httpsdoiorg1023073235759

[10] N Barbier P Couteron R Lefever V Deblauweand O Lejeune Spatial decoupling of facilitationand competition at the origin of gapped vegetationpatterns Ecology 891521ndash1531 2008 URL httpswwwncbinlmnihgovpubmed18589517

[11] M Baudena F DrsquoAndrea and A Provenzale Anidealized model for tree-grass coexistence in savannasthe role of life stage structure and fire disturbances JEcol 9874ndash80 2010 URL httpdxdoiorg101111j1365-2745200901588x

[12] M Baudena SC Dekker PM Van BodegomB Cuesta SI Higgins V Lehsten CH ReickM Rietkerk S Scheiter Z Yin MA Zavala andV Brovkin Forest savannas and grasslands bridgingthe knowledge gap between ecology and dynamicglobal vegetation models Biogeosciences Discuss119471ndash9510 2014 URL httpdxdoiorg105194bgd-11-9471-2014

[13] B Beckage LJ Gross and WJ Platt Grass feed-backs on fire stabilize savannas Ecol Modell2222227ndash2233 2011 URL httpdxdoiorg101016jecolmodel201101015

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[17] WJ Bond What limits trees in C4 grasslands andsavannas Annu Rev Ecol Evol Syst 39641ndash6592008 URL httpdxdoiorg101146annurevecolsys39110707173411

[18] WJ Bond and JE Keeley Fire as a global rdquoherbi-vorerdquothe ecology and evolution of flammable ecosys-tems Trends in Ecoloy and Evolution 20(7)387ndash3942005 URL httpdxdoiorg101016jtree200504025

[19] WJ Bond and GF Midgley A proposed CO2-controlled mechanism of woody plant invasion ingrasslands and savannas Global Change Biol 6865ndash869 2000 URL httpsdoiorg101046j1365-2486200000365x

[20] WJ Bond GF Midgley and FI Woodward Whatcontrols south african vegetation-climate or fire SAfr J Bot 6979ndash91 2003 URL httpsdoiorg101016S0254-6299(15)30362-8

[21] WJ Bond and C J Parr Beyond the forest edge Ecol-ogy diversity and conservation of the grassy biomesBiol Conserv 1432395ndash2404 2010 URL httpdxdoiorg101016jbiocon200912012

[22] WJ Bond FI Woodward and GF Midgley Theglobal distribution of ecosystems in a world withoutfire New Phytol 165525ndash538 2005 URL httpdxdoiorg101111j1469-8137200401252x

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[24] A Bouvet S Mermoz Thuy Le Toan L VillardR Mathieu L Naidoo and GP Asner An above-ground biomass map of african savannahs and wood-lands at 25m resolution derived from alos palsarRemote Sensing of Environment 206156 ndash 173 2018URL httpsdoiorg101016jrse201712030

[25] J Brookman-Amissah JB Hall Swaine MD andJY Attakorah A re-assessment of a fire protectionexperiment in north-eastern Ghana savanna Journalof Applied Ecology 17(1)85ndash99 1980 URL httpwwwjstororgstable2402965

[26] G Bucini and N P Hanan A continental-scaleanalysis of tree cover in african savannas Global Ecol

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[28] C De Michele F Accatino R Vezzoli and RJScholes Savanna domain in the herbivores-fire param-eter space exploiting a tree-grass-soil water dynamicmodel J Theor Biol 289(0)74ndash82 2011 URLhttpdxdoiorg101016jjtbi201108014

[29] A Dhooge W Govaerts and Yu A Kuznetsov MAT-CONT A MATLAB package for numerical bifurcationanalysis of ODEs ACM TOMS 29(2)141ndash164 2003URL httpdxdoiorg101145779359779362

[30] F Dietrich G Koster M Seitz and I von SiversBridging the gap From cellular automata to differentialequation models for pedestrian dynamics Journal ofComputational Science 5(5)841 ndash 846 2014 URLhttpsdoiorg101016jjocs201406005

[31] A Diouf N Barbier AM Lykke P Couteron V De-blauwe A Mahamane M Saadou and J BogaertRelationships between fire history edaphic factor andwoody vegetation structure and composition in a semi-arid savanna landscape (niger west africa) Appl VegSci 15488ndash500 2012 URL httpsdoiorg101111j1654-109X201201187x

[32] P Drsquoodorico F Laio and L Ridolfi A probabilis-tic analysis of fire-induced tree-grass coexistence insavannas Am Nat 167(3)E79ndashE87 2006 URLhttpdxdoiorgdoi101086500617

[33] J Dohn F Dembele M Karembe Aristides Mous-takas Kosiwa A Amevor and Niall P Hanan Treeeffects on grass growth in savannas competition facil-itation and the stress-gradient hypothesis Journal ofEcology 101(1)202ndash209 2012 URL httpdxdoiorg1011111365-274512010

[34] Y Dumont J-C Soulie and F Michel Model-ing oil palm pollinator dynamics using deterministicand agent-based approaches applications on fruit setestimates some preliminary results MathematicalMethods in the Applied Sciences 0(0) URL httpsdoiorg101002mma4858

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[36] C Favier J Aleman L Bremond M A DuboisV Freycon and J M Yangakola Abrupt shifts inafrican savanna tree cover along a climatic gradientGlobal Ecol Biogeogr 21787ndash797 2012 URLhttpdxdoiorg101111j1466-8238201100725x

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[44] N Govender WSW Trollope and BW Van WilgenThe effect of fire season fire frequency rainfalland management on fire intensity in savanna vegeta-tion in south africa J Appl Ecol 43(4)748ndash7582006 URL httpdxdoiorg101111j1365-2664200601184x

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partial differential equations Physica D NonlinearPhenomena 10(1)128 ndash 134 1984 URL httpsdoiorg1010160167-2789(84)90255-0

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[77] S Scheiter and SI Higgins Partitioning of root andshoot competition and the stability of savannas AmNat 179587ndash601 2007 URL httpdxdoiorg101086521317

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[83] AC Staver S Archibald and S Levin Tree cover insub-saharan Africa Rainfall and fire constrain forestand savanna as alternative stable states Ecology92(5)1063ndash1072 2011 URL httpdxdoiorg10189010-16841

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dynamics Ecol Modell 30411ndash21 2015 URL httpdxdoiorg101016jecolmodel201502015

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[87] A Tchuinte Tamen Study of a Generic Mathemat-ical Model of Forest-Savanna Interactions Case ofCameroon PhD thesis University of Yaoune I 2017

[88] A Tchuinte Tamen P Couteron and Y Dumont Ef-fect of fire frequency and rainfall on treendashgrass dynam-ics Capturing the forestndashsavanna distributions at bio-geographic scale ArXiv preprint arXiv1802059862018 URL httpsarxivorgabs180205986

[89] A Tchuinte Tamen Y Dumont J J Tewa S Bowongand P Couteron Tree-grass interaction dynamics andpulsed fires mathematical and numerical studies ApplMath Mod 40(11-12)6165ndash6197 June 2016 URLhttpsdoiorg101016japm201601019

[90] A Tchuinte Tamen Y Dumont J J Tewa S Bowongand P Couteron A minimalistic model of tree-grassinteractions using impulsive differential equations andnon-linear feedback functions of grass biomass ontofire-induced tree mortality Math Comput Simul133265ndash297 March 2017a URL httpdxdoiorg101016jmatcom201603008

[91] A Tchuinte Tamen J J Tewa P CouteronS Bowong and Y Dumont A generic modeling offire impact in a tree-grass savanna model Biomath31407191 2014 URL httpdxdoiorg1011145jbiomath201407191

[92] JJ Tewa IV Yatat Djeumen and S BowongPredator-prey model with holling response function oftype ii and sis infectious disease Appl Math Modell37(7)4825 ndash 4841 2013 URL httpsdoiorg101016japm201210003

[93] K Thonicke S Venevsky S Sitch and W CramerThe role of fire disturbance for global vegetationdynamics coupling fire into a dynamic global vege-tation model Global Ecol Biogeogr 10(6)661ndash6772001 URL httpsdoiorg101046j1466-822X200100175x

[94] D Tilman Competition and biodiversity in spatiallystructured habitats Ecology 75(1)2ndash16 1994 URLhttpswwwjstororgstable1939377

[95] M Torello-Raventos TR Feldpausch K VeenendaalF Schrodt G Saiz TF Domingues G DjagbleteyA Ford J Kemp BS Marimon BH MarimonE Lenza JA Ratter L Maracahipes D SasakiB Sonke L Zapfack H Taedoumg D VillarroelM Schwarz CA Quesada F Yoko Ishida GBNardoto K Affum-Baffoe L Arroyo D BowmanH Compaore K Davies A Diallo NM FyllasM Gilpin F Hien M Johnson TJ Killeen D Met-calfe HS Miranda M Steininger J ThomsonK Sykora E Mougin P Hiernaux MI Bird J Grace

SL Lewis OL Phillips and J Lloyd On thedelineation of tropical vegetation types with an em-phasis on forestsavanna transitions Plant Ecology ampDiversity 6(1)101ndash137 2013 URL httpsdoiorg101080175508742012762812

[96] JD Touboul C Staver and SA Levin On thecomplex dynamics of savanna landscapes Proceedingsof the National Academy of Sciences 2018 URLhttpsdoiorg101073pnas1712356115

[97] WSW Trollope Fire in savanna In P de V Booysenand NM Tainton editors Ecological Effects of Firein South African Ecosystems pages 149ndash175 Springer-Verlag Berlin-Heidelberg-New York-Tokyo 1984

[98] WSW Trollope Personal perspectives on commercialversus communal african fire paradigms when usingfire to manage rangelands for domestic livestock andwildlife in southern and east african ecosystems FireEcology 7(1)57ndash73 2011 URL httpsdoiorg104996fireecology0701057

[99] WSW Trollope and LA Trollope Fire in africansavanna and other grazing ecosystems paper presentedat the seminar on rsquoforest fire and global changersquo heldin shshenkoye in the russian federation 4-10 August1996

[100] WSW Trollope and LA Trollope Fire effectsand management in african grasslands and savan-nas Range and Animal Sci Resour Manag 2121ndash145 2010 URL wwweolssnetsample-chaptersc10E5-35-18pdf

[101] F Van Langevelde C Van de Vijver L KumarJ Van de Koppel N De Rider and J Van AndelEffects of fire and herbivory on the stability of savannaecosystems Ecology 84(2)337ndash350 2003 URLhttpdxdoiorg1018900012-9658(2003)084[0337EOFAHO]20CO2

[102] BW Van Wilgen N Govender HC Biggs D Ntsalaand XN Funda Response of savanna fire regimes tochanging fire-management policies in a large africannational park Conserv Biol 181537ndash1540 2004URL httpsdoiorg101111j1523-1739200400362x

[103] B Walker D Ludwig CS Holling and RM Peter-man Stability of semi-arid savanna grazing systemsJournal of Ecology 69(2)473ndash498 1981 URL httpswwwjstororgstablepdf2259679pdf

[104] B Walker and I Noy-Meir Aspects of the stabilityand resilience of savanna ecosystems In BJ Huntley

and BH Walker editors Ecology of tropical savan-nas pages 555ndash590 Springer-Verlag Berlin Germany1982

[105] H Walter Ecology of tropical and subtropical vegeta-tion Oliver and Boyd Edinburgh UK 1971

[106] JR Weimar Cellular automata for reaction-diffusionsystems Parallel Computing 23(11)1699 ndash 17151997 Cellular automata URL httpsdoiorg101016S0167-8191(97)00081-1

[107] JF Weltzin and MB Coughenour Savanna treeinfluence on understory vegetation and soil nutrientsin northwestern Kenya J Veg Sci 1325ndash334 1990URL httpsdoiorg1023073235707

[108] S Wolfram Twenty problems in the theory of cellularautomata Physica Scripta 1985(T9)170 1985 URLhttpstacksioporg1402-48961985i=T9a=029

[109] V Yatat P Couteron and Y Dumont Spatiallyexplicit modelling of tree-grass interactions in fire-prone savannas a partial differential equations frame-work Ecol Compl pages 290ndash313 2018 URLhttpsdoiorg101016jecocom201706004

[110] V Yatat P Couteron J J Tewa S Bowong andY Dumont An impulsive modelling framework offire occurrence in a size-structured model of treendashgrass interactions for savanna ecosystems J MathBiol 74(6)1425ndash1482 2017 URL httpdxdoiorg101007s00285-016-1060-y

[111] V Yatat and Y Dumont FKPP equation with impulseson unbounded domain In Mathematical Methods andModels in Biosciences 2018 URL httpdxdoiorg1011145texts201711157

[112] V Yatat Y Dumont J J Tewa P Couteron andS Bowong Mathematical analysis of a size structuredtree-grass competition model for savanna ecosystemsBiomath 31404212 2014 URL httpdxdoiorg1011145jbiomath201404212

[113] IV Yatat Djeumen Mathematical analysis of size-structured tree-grass interactions models for savannaecosystems PhD thesis University of Yaounde I 2018

[114] J Youta Happi Arbres contre graminees la lente in-vasion de la savane par la foret au Centre-CamerounPhD thesis Universite de Paris IV 1998

[115] K Yu and P Drsquoodorico An eco-hydrological frame-work for grass displacement by woody plants in savan-nas J Geophys Res Biogeosci 119192ndash206 2014 URL httpdxdoiorg1010022013JG002577

APPENDIX

Table V Parameter values used to obtain Figure 7-(a) page 17 From Accatino et al (2010) [4] (here see model(3))

w1 minus ε yrminus1 cT yrminus1 cG yrminus1 γT yrminus1 γG yrminus1 δT yrminus1 δG yrminus1 δF minus

345 20 30 10 2 180 004 28 035

Table VI Parameter values used to get Figure 7-(b) page 17 From Tchuinte Tamen et al (2018) [88] (here seemodel (6))

cG thaminus1 cT thaminus1 bG mmyrminus1 bT mmyrminus1 aG yrminus1 aT yrminus1

20 450 501 1192 00029 00045

dG minus dT minus γG yrminus1 γT yrminus1 δG yrminus1 δT yrminus1

1473 1067 25 1 001 01

λfG minus λminfT minus λmax

fT minus p tminus1ha α thaminus1 ηTG hatminus1yrminus1

03 005 07 001 1 001

Introduction
A brief review on space-implicit tree-grass interactions modelling
- Tree-grass coexistence and possible alternative stable states
- Lines of thoughts
- - Main published modelling options
  - Reaching sensible predictions from minimal models
  - - A seminal big picture at biogeographic scale
    - Fire frequency grass biomass and fire impact
    - Tree survival
    - Relating to water resource
    - Impulsive time-periodic occurrences of fire events
    - - Discussion and prospects
      - Conclusion
      - References

Page 2: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

I INTRODUCTION

Savannas have been identified by biogeogra-phers as a biome corresponding to warm meanannual temperatures (gt 20degC) and a broad rangeof intermediate mean annual rainfall (100ndash2000mmyrminus1) (Sarmiento (1984) [76] Youta Happi(1998) [114] Abbadie et al (2006) [1] Lehmannet al (2011) [58]) Such climatic context predom-inates along rainfall gradients leading from sub-equatorial wet climates to hot arid climates Thewider definition to which we refer here tends tointegrate climatic variant allowing for nearly desertvegetation or on the contrary seasonal tropicalforests Savannas display specific interplays ofnatural constraints that prevent or at least impedeclosure of woody cover and ensuing suppressionof light-demanding herbs and grasses A centralalbeit non-exclusive cause for this is the rsquoideal fireclimatersquo (Trollope (2011) [98]) that characterizestropical regions with seasonal droughts alternatingwith warm and wet rainy seasons producing highherbaceous biomass that once dried-up becomeshighly ignitable and fuels fires (Frost et al (1986)[39] Thonicke et al (2001) [93] Govender et al(2006) [44]) High frequency of lightning stormswhich is a characteristic of Africa (Abbadie et al(2006) [1] Trollope (2011) [98]) also contributesto make it the rdquoFire continentrdquo even though presentfire regimes mostly rely on human-made ignitions(Archibald et al (2009) [7] Govender et al (2006)[44] Trollope (2011) [98])

Dynamics of vegetation within the savannabiome has long interested ecologists as it clearlydeparts from the classical post-disturbance succes-sion pathways that are expected to rapidly bringback closed canopy forest as observed in most oftemperate and wet tropical climatic zones (Bond etal (2005) [22]) The last decades have witnessedburgeoning efforts of modelling as to account forthe possibly long-lasting coexistence of grassy andwoody components and try to predict potentialshifts from two-phased vegetation physiognomiesThese efforts have however not yet delivered anintegrated picture liable to furnish at broad scales(ie for fractions of continents) sensible predic-

tions of possible vegetation dynamics Such a bigpicture is nevertheless desirable for figuring outthe future of vegetation in the face of climate andanthropic change scenarios (Mayaux et al (2004)[60] Bond et al (2005) [22] Archibald et al(2009) [7] Accatino et al (2010) [4] Favier et al(2012) [36]) It is also necessary for applicationsto territories devoid of reference data and long-term observation sites as it is the case for most oftropical Africa

The objectives of the present contribution arefourfold It first aims at recalling and synthetizingthe main array of facts about ecological processesand resulting vegetation dynamics that modelsshould aim to capture and render (see section II)Second in order to claim genericity we synthetizethe main modelling options present in the liter-ature and put emphasis on minimalistic modelscapturing only essential processes while retainingsufficient mathematical tractability and restrictingthemselves to a minimal set of assessable pa-rameters (see section III) Thirdly on this basiswe argue that such models have now becomemore comprehensive and useful for meaningfulpredictions (see section IV) Finally we discusshow those models may now help guiding datacollection for improved calibration and testing ofdynamical hypotheses (see section V)

II A BRIEF REVIEW ON SPACE-IMPLICIT

TREE-GRASS INTERACTIONS MODELLING

A Tree-grass coexistence and possible alternativestable states

Over very large tropical territories field ob-servers have documented long lasting coexistenceof notable levels of grass and woody biomass(Backeus (1992) [9]) The most frequently re-ported form of coexistence is observed locallythrough vegetation physiognomies that associatefairly continuous grassy cover and more or lessscattered populations of trees and shrubs of vary-ing clumping levels This is referred to as sa-vanna physiognomy (see Figure 1) Such vege-tation types mixing both lifeforms are manifoldand progressively merge in space or through time

B Lines of thoughts

dT

dG

(1)

dT

dS

dt=

p

minusτGSG

dT

dG

(3)

wherep

ω(G) =Gθ

Gθ + αθ (4)

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 3: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

B Lines of thoughts

dT

dG

(1)

dT

dS

dt=

p

minusτGSG

dT

dG

(3)

wherep

ω(G) =Gθ

Gθ + αθ (4)

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 4: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

dT

dG

(1)

dT

dS

dt=

p

minusτGSG

dT

dG

(3)

wherep

ω(G) =Gθ

Gθ + αθ (4)

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 5: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

dT

dG

(1)

dT

dS

dt=

p

minusτGSG

dT

dG

(3)

wherep

ω(G) =Gθ

Gθ + αθ (4)

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 6: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

dT

dS

dt=

p

minusτGSG

dT

dG

(3)

wherep

ω(G) =Gθ

Gθ + αθ (4)

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 7: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 8: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 9: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

Yatat et al (2014) [112] X X

Yatat et al (2017) [110] X X

Yatat et al (2018) [109] X X

MINIMAL MODELS

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 10: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

0

10

20

30

40

50

60

70

80

90

100

TO

PK

ILL

minus

46

8

1316

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT minus m

TO

PK

ILL

minus

05 1 2 3 4 5

9

2729

43

49

69

75

8991

9999

59

(b)

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 11: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 12: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

0 2 4 6 8 100

2

4

6

8

10

Fire

impa

ct (

w(G

))

Holling type I

(a)

0 2 4 6 8 100

02

04

06

08

1

Fire

impa

ct (

w(G

))

Holling type II

Holling type III

(b)

equilibria

monostability

2 savannas

bistability

Hollingtype III

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 13: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 14: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

C Tree survival

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 15: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

dG

dt= rG(W )G

(1minus G

KG(W )

)minus δGG

dT

dt= rT (W )T

(1minus T

KT (W )

)minus δTT

minusfϑ(T )ω(G)T

G(0) =G0 T (0) = T0

(6)

bull rG(W ) =γGW

bG +Wand rT (W ) =

γTW

bull KG(W ) =cG

cT1 + dT eminusaTW

fire return period

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 16: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

dG

dt= rG(W )G

(1minus G

KG(W )

)minusδGG

minusηTGTG

dT

dt= rT (W )T

(1minus T

KT (W )

)minusδTT

t 6= nτ

n = 1 2 Nf

(7)

π(nτ + θ)

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 17: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

(a)

(b)

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 18: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

Ranges References

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 19: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=051)

(b)

0 5 10 15 20

Grass biomass (tha)

0

10

20

30

40

50

60

70

80

90

100

Tre

e b

iom

ass

(th

a)

(W=946 f=07)

(c)

(a) (b) (c)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=04)

(a)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=05)

(b)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=056)

(c)

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

Page 20: A tribute to the use of minimalistic spatially-implicit ...€¦ · I.V. Yatat Djeumen, A. Tchuint´e Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ... I. INTRODUCTION

0 5 10 15 20

Grass biomass (tha)

0

20

40

60

80

100

120

140

160

180

200

Tre

e b

iom

ass

(th

a)

(W=1008 f=03)

(a) (b)

(c) (d)

VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

345 20 30 10 2 180 004 28 035

20 450 501 1192 00029 00045

1473 1067 25 1 001 01

03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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VI CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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ACKNOWLEDGEMENTS

REFERENCES

APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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APPENDIX

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03 005 07 001 1 001

Introduction
- Lines of thoughts
    - Tree survival
      - Conclusion
      - References

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