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Successional dynamics in Neotropical forests are asuncertain as
they are predictableNatalia Nordena,b,1, Hctor A. Angaritab, Frans
Bongersc, Miguel Martnez-Ramosd, Iigo Granzow-de la Cerdae,Michiel
van Breugelf,g, Edwin Lebrija-Trejosg,h, Jorge A. Meavei, John
Vandermeerj, G. Bruce Williamsonk,Bryan Fineganl, Rita Mesquitam,
and Robin L. Chazdonn
aFundacin Cedrela, Bogot 111311, Colombia; bDepartamento de
Ecologa y Territorio, Pontificia Universidad Javeriana, Bogot
110231, Colombia; cForestEcology and Forest Management Group,
Department of Environmental Sciences, Wageningen University, 6700
AAWageningen, The Netherlands; dInstitutode Investigaciones en
Ecosistemas y Sustentabilidad, Universidad Nacional Autnoma de
Mxico, Morelia 58190, Michoacn, Mexico; eDepartamento deBiologa
Animal, Biologa Vegetal y Ecologa, Universidad Autnoma de
Barcelona, E-08193 Bellaterra, Spain; fYaleNational University of
SingaporeCollege, Singapore 138614; gSmithsonian Tropical Research
Institute, Apartado 0843-03092, Balboa, Panama; hDepartment of
Forest Resources, Universityof Minnesota, St. Paul, MN 55108;
iFacultad de Ciencias, Departamento de Ecologa y Recursos
Naturales, Universidad Nacional Autnoma de Mxico,Mxico 04510, DF,
Mexico; jDepartment of Ecology and Evolutionary Biology, University
of Michigan, Ann Arbor, MI 48109; kDepartment of
BiologicalSciences, Louisiana State University, Baton Rouge, LA
70808; lProduction and Conservation in Forests Program, Tropical
Agricultural Centre for Research andHigher Education, Apartado
93-7170, Turrialba, Costa Rica; mBiological Dynamics of Forest
Fragments Project, Instituto Nacional de Pesquisas da
Amazonia,Manaus, AM 69011-970, Brazil; and nDepartment of Ecology
and Evolutionary Biology, University of Connecticut, Storrs, CT
06269-3043
Edited by William J. Bond, University of Cape Town, Cape Town,
South Africa, and approved May 20, 2015 (received for review
January 8, 2015)
Although forest succession has traditionally been approached as
adeterministic process, successional trajectories of vegetation
changevary widely, even among nearby stands with similar
environmentalconditions and disturbance histories. Here, we provide
the firstattempt, to our knowledge, to quantify predictability and
uncertaintyduring succession based on the most extensive long-term
datasetsever assembled for Neotropical forests. We develop a novel
approachthat integrates deterministic and stochastic components
into differentcandidate models describing the dynamical
interactions among threewidely used and interrelated forest
attributesstem density, basalarea, and species density. Within each
of the seven study sites, suc-cessional trajectories were highly
idiosyncratic, even when controllingfor prior land use,
environment, and initial conditions in these attrib-utes. Plot
factors were far more important than stand age in explain-ing
successional trajectories. For each site, the best-fit model was
ableto capture the complete set of time series in certain
attributes onlywhen both the deterministic and stochastic
components were set tosimilar magnitudes. Surprisingly,
predictability of stem density, basalarea, and species density did
not show consistent trends across at-tributes, study sites, or land
use history, and was independent of plotsize and time series
length. The model developed here represents thebest approach, to
date, for characterizing autogenic successional dy-namics and
demonstrates the low predictability of successional tra-jectories.
These high levels of uncertainty suggest that the impacts
ofallogenic factors on rates of change during tropical forest
successionare far more pervasive than previously thought,
challenging the wayecologists view and investigate forest
regeneration.
dynamical models | predictability | succession | tropical
secondary forests |uncertainty
Unexplained variation (uncertainty) is ubiquitous in ecology,and
often constrains our ability to elucidate the mechanismsthat drive
variation in forest structure and dynamics. This issue isreflected
in the long-standing controversy over the relative impor-tance of
determinism and stochasticity in shaping community as-sembly (14).
Although it has been widely demonstrated that bothdeterministic and
stochastic processes drive community assembly inmature forests (5,
6), their relative importance in explaining forestsuccession has
not been rigorously evaluated (7). More than one-halfof the
tropical biome is in some stage of recovery from past
humandisturbance (8), yet no previous study has quantitatively
assessed theextent to which regenerating forests follow predictable
trajectories.Since the early days of community ecology, succession
has been
viewed either as a deterministic (1) or a stochastic (2)
process.Forest succession, however, has been traditionally
approached as apredictable process, mostly driven by autogenic
factors intrinsic tothe forest site (9, 10). Deviations from this
expectation are usually
attributed to allogenic factors, such as prior land use or
priorityeffects (11, 12). As a result, most of our knowledge on
forestsuccession is based on chronosequences (13), a
space-for-timesubstitution approach that assumes that succession
follows asingle, largely deterministic trajectory over time. Recent
studies,however, have shown that successional pathways vary widely,
evenamong neighboring stands with similar environmental
conditionsand disturbance history (1418). In the case of
posthurricanesuccession in Nicaragua, such variation has been
attributed tostochastic processes associated with nonequilibrium
communitydynamics (18). As long-term successional studies in the
tropics arerare, assessing predictability of successional
trajectories in species-rich communities has not been possible
across a broader range ofgeographical and historical
settings.Successional dynamics has been typically studied through
the
lens of three widely used forest attributesstem density,
basalarea, and species density, whose dynamics are often
evaluatedindependently of one another (12). These metrics, however,
arelikely to change interdependently during succession. In
partic-ular, successional changes in stem density are associated
withchanges in basal area, and vice versa (9, 17). Yet, other
possible
Significance
Although forest succession has been approached as a predict-able
process, successional trajectories vary widely, even amongnearby
stands with similar environmental conditions and dis-turbance
histories. We quantified predictability and uncertaintyduring
tropical forest succession using dynamical models de-scribing the
interactions among stem density, basal area, andspecies density
over time. We showed that the trajectories ofthese forest
attributes were poorly predicted by stand age andvaried
significantly within and among sites. Our models repro-duced the
general successional trends observed, but high levelsof noise were
needed to increase model predictability. Theselevels of uncertainty
call into question the premise that suc-cessional processes are
consistent over space and time, andchallenge the way ecologists
view tropical forest regeneration.
Author contributions: F.B., M.M.-R., M.v.B., E.L.-T., J.A.M.,
J.V., G.B.W., B.F., R.M., andR.L.C. designed research; F.B.,
M.M.-R., I.G.-d.l.C., M.v.B., E.L.-T., J.A.M., J.V., G.B.W.,
B.F.,R.M., and R.L.C. performed research; H.A.A. contributed new
reagents/analytic tools; N.N.and H.A.A. analyzed data; N.N. and
R.L.C. wrote the paper, and H.A.A., F.B., M.M.-R.,
I.G.-d.l.C.,M.v.B., E.L.-T., J.A.M., J.V., G.B.W., B.F., and R.M.
assisted with writing the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.1To whom correspondence
should be addressed. Email: [email protected].
This article contains supporting information online at
www.pnas.org/lookup/suppl/doi:10.1073/pnas.1500403112/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1500403112 PNAS | June 30,
2015 | vol. 112 | no. 26 | 80138018
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interrelations between these attributes have not been
previouslyexplored in a successional scenario. For instance,
although rates ofchange in species density are expected to depend
upon changes instem density (19, 20), it is not clear whether
changes in basal areaaffect rates of species gain or loss. Also,
the causal relationshipbetween species density and rates of change
in stem density andbasal area is poorly understood. To our
knowledge, a clear syn-thesis addressing the simultaneous
interdependence of the rates ofchange of these three forest
attributes is currently lacking. Apromising approach to address
this issue is to view regeneratingtropical forests as complex
adaptive systems, which integrate manyof the features
characterizing reassembling plant communities,namely
self-organization, memory, nonlinearity, and uncertainty(21, 22).
Through such a holistic perspective, we can gain a mech-anistic
understanding of how the interacting components influ-encing
succession produce a system dynamics that cannot be easilypredicted
from their individual behavior (23).Here, we develop a novel
modeling approach that addresses
these dynamic interdependencies and that quantifies
predictabilityand uncertainty in successional pathways by
integrating both de-terministic and stochastic components. We apply
these models toan unparalleled dataset from seven lowland tropical
secondaryforests spanning four Neotropical countries (Brazil, Costa
Rica,Mexico, and Nicaragua). Each study site includes 415 plots
thatdocument long-term successional forest dynamics (Table
S1).Within each site, plots were established in close proximity
andshare similar land use history, climate, and soil conditions.
Thesedata comprise most of the studies on secondary forest dynamics
inthe Neotropics and encompass different land use histories
andclimate regimes, providing an unprecedented opportunity to
in-vestigate the generality of the successional dynamics
observed.To quantify predictability and uncertainty during
succession, we
first illustrate among-plot variability in the successional
trajectoriesof stem density, basal area, and species density, and
evaluate theeffect of stand age on these forest attributes. Then,
we quantifythe predictability of successional trajectories within
each site bymodeling succession as the realization of a dynamical,
strictly de-terministic process resulting from the initial
conditions in stemdensity, basal area, and species density, and the
simultaneous in-teraction among these state variables over
successive time steps(Fig. 1). Finally, we assess the degree of
uncertainty underlying thepredictability of the deterministic model
by incorporating a sto-chastic component governed by a parameter
that defines the rel-ative magnitude of the deterministic and the
stochastic components(Fig. 1). This approach allows us to address
the following questions:(i) How much of the variation in
successional trajectories withineach site is explained by stand
age? (ii) Can a single dynamicalmodel describe the simultaneous
interaction among rates of changein stem density, basal area, and
species density? (iii) What is therelative importance of
predictability and uncertainty in successionaltrajectories within
each site? (iv) Is any of the forest attributes morepredictable
than the others? (v) Is the degree of uncertainty insuccessional
trajectories related to previous land use?
ResultsVariability in Successional Trajectories and Effect of
Stand Age.Successional trajectories of stem density, basal area,
and spe-cies density varied widely within and among sites (Fig. 2).
Withineach site, plot identity (random effect) was more important
thanstand age since abandonment (fixed effect) in explaining
varia-tion in forest attributes, accounting for over 60% of the
totalvariance in most cases (Table S2). For instance, in Brazil 1,
plotidentity explained over 80% of the variance in all three
attrib-utes, and stand age did not significantly predict basal area
orspecies density. Sites without previous land use (Costa Rica 2and
Nicaragua) showed similar patterns to sites used previouslyas
pastures (Brazil 2 and Costa Rica 1), with plot identityexplaining
over 90% of the variance in certain cases. Strikingly,the
contribution of age and its interaction with plot identity waslow
in comparison with that of plot identity alone, explaining lessthan
20% of the total variance within sites (Table S2).
Predictability in Successional Trajectories. To quantify
predictabilityof successional trajectories within each site, we
modeled suc-cession as the realization of a dynamical process
resulting frominitial conditions in stem density, basal area, and
species density,and their simultaneous interaction, over successive
time steps.We evaluated the fit of three candidate dynamical
models(Methods). The nonlinear dynamical model performed
markedlybetter than the linear model or the linear model with
in-teractions, both of which performed poorly (Table S3).As the
nonlinear model included nine terms and 18 parameters
(Methods), we attempted to reduce the number of
parametersthrough backward elimination of terms that were not
supported byavailable data (9). More specifically, rates of change
in stemdensity or in basal area are not likely to be causally
related withspecies density, and the relationship between rates of
change inspecies density and basal area has not been explored yet
in asuccessional scenario. We tested the seven possible
combinationsthat had eight, seven, or six terms (16, 14, and 12
parameters,respectively) instead of the nine terms (18 parameters)
included inthe full nonlinear model. The model that best fitted the
observeddata differed among sites (Table S4), indicating that the
processesdriving successional dynamics were neither consistent nor
uniformacross a broad range of secondary forests.Fig. 3 shows how
the derivatives (rates of change) of stem density,
basal area, and species density varied as a function of each
attributealone, based upon the fitted parameters of the best-fit
nonlinearmodel for each site. Some bivariate patterns were
consistent amongsites, suggesting generality in successional
processes across studysites. In most sites, increasing stem density
led to decreasing rates ofchange in stem density, but to positive
and increasing rates ofchange in basal area (Fig. 3 A and D). These
results suggest hightree mortality in dense early successional
stands, whereas theremaining trees rapidly accumulate basal area.
Similarly, stands withhigh basal area showed positive and
increasing rates of change instem density in most sites, but
negative and decreasing rates ofchange in basal area (Fig. 3 B and
E). These findings indicate that,when stands reach a saturation
point in terms of basal area, regu-lation occurs through the death
of large trees, rather than throughrecruitment limitation. Also, in
most sites, rates of change in speciesdensity increased as stem
density increased (Fig. 3G), but decreasedas species density
increased (Fig. 3I). These results reflect speciescolonization
through tree recruitment early in succession, whereas,
X(t) = {D(t), BA(t), S(t)}
Spec
ies
dens
ity
Basal area Stem density
Deterministic trajectory:
g(X(t))
g(X(t) + h(X(t))
System state at time t
Probability distribution of X at time t + t
Fig. 1. Illustration of the model as the realization of the
process X(t) starting attwo different initial conditions. The model
integrates predictability and un-certainty as described by a system
of stochastic differential equations, where Xdenotes the state of
the system at a given time. The system is characterized bythree
state variables: stem density, basal area, and species density. The
thick linesrepresent two trajectories, expressed by the
deterministic component of themodel only, in a phase space defined
by normalized stem density, basal area, andspecies density, and
starting at two different initial conditions. The thin, dottedlines
represent two trajectories, as expressed by the stochastic model.
As thezoom shows, the stochastic trajectory is the result of a
random walk starting atthe initial condition. The stochastic model
drives the system from time t to timet +t, and the possible
outcomes follow a Gaussian probability distribution.
8014 | www.pnas.org/cgi/doi/10.1073/pnas.1500403112 Norden et
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as succession unfolds, rates of species gain reach a saturation
pointdetermined by the number of species that can establish in
thecommunity (20). Notably, rates of change in stem density,
basalarea, and species density showed wide variation among sites,
even incases when the direction of these trends was consistent
among sites(Fig. 3 and Table S5).The dynamical models also revealed
several interactions among
state variables that have not been demonstrated in previous
stud-ies. In Brazil 1 and Mexico dry, increased species density was
as-sociated with positive and increasing rates of change in
stemdensity (Fig. 3C). In Brazil 2 and Mexico wet, rates of change
inbasal area decreased dramatically at low levels of species
density,and then stabilized (Fig. 3F). Although these patterns may
notreflect causal relationships, they might mirror the resultant
pat-terns of underlying processes specific to some sites. For
instance,during the first years of succession in Mexico wet, the
low diversityassemblage of pioneer species experienced an acute
mortalityepisode that was accompanied by a sudden reduction in
basalarea (17).
Relative Importance of Predictability and Uncertainty in
SuccessionalTrajectories.Despite the overall good fit of the
nonlinear deterministicmodels (Fig. S1), the correlation between
observed and predictedtemporal trajectories of the three state
variables for each plot withineach site was not significantly
positive in many cases (143/201; TableS6). To assess the degree of
uncertainty underlying the pre-dictability of the deterministic
model, we incorporated a stochasticcomponent to the nonlinear
models by generating 1,000 trajectoriesgiven the observed initial
conditions and a parameter , whichmodulates the amount of noise
integrated to the model. The modelenvelopes defined by the
stochastic trajectories included the great-est part of the observed
trajectories only when the relative magni-tude of the stochastic
and deterministic components was set to besimilar (0.8 < <
1.2; Fig. 4), and only in one or two of the threeattributes. The
sole site for which the model was able to predict theentire set of
time series for all three attributes was Mexico wet, andthis needed
an important contribution of the stochastic component(> 1).
Although the predictability of the model (the fraction ofdata
points within the model envelope) increased as the
relativemagnitude of the stochastic component increased, it never
reachedthe maximum value (one) for many forest attributes in
certain sites(e.g., Costa Rica 1 and Mexico dry). Indeed, in some
cases, thepredictability of the stochastic model decreased as
increased (e.g.,species density in Nicaragua) because the addition
of noise to thenonlinear deterministic model launched the
trajectory of the systeminto a region of phase space with negative
values in the state vari-ables, resulting in a mathematical
artifact (Methods).Overall, the observed patterns were highly
idiosyncratic, and the
predictability of forest attributes did not show consistent
trendsacross attributes, sites, or land use history. None of the
forests at-tributes showed higher predictability than the others
(Fig. 4), andsites without any previous land use (Brazil 1, Costa
Rica 2, andNicaragua) showed similar patterns to those used for
pastures(Brazil 2 and Costa Rica 1) or agriculture (Mexico wet and
Mexicodry). Furthermore, the magnitude of noise intensity required
toincrease model predictability was independent of plot size and
oftime series length (Figs. S2 and S3). Indeed, the larger Costa
Ricanplots (>1 ha) showed similar idiosyncratic patterns to
those ob-served in other sites.Age
1200
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1lizarBStem density Basal area Species density
Fig. 2. Observed successional trajectories of stem density,
basal area, andspecies density. For each plot within each site,
temporal changes in stemdensity, basal area (in square meters), and
species density are plotted againstage since pasture or clearcut
abandonment (lines connecting same symbols).Each of these
attributes was standardized by plot size, depending upon thesite
(Methods).
-1
-0.5
0
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0 0.5
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axst
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ax b
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dens
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0 0.5 1Brazil 1Brazil 2Costa Rica 1Costa Rica 2Mexico wetMexico
dryNicaragua-1
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A B C
D E F
G H I
Fig. 3. Among-site variation in the values of the fitted
parameters of thedeterministic model. Derivatives of stem density
(AC), basal area (DF), andspecies density (GI) as a function of
each of these properties alone, obtainedfrom the fitted parameters
of the nonlinear dynamical model in Eq. 2c. Thederivative of each
forest attribute is divided by its observed maximum toevaluate the
contribution of each attribute to each derivative on a
unitlessscale. Note that stand age is not explicitly addressed in
the plot axes.
Norden et al. PNAS | June 30, 2015 | vol. 112 | no. 26 |
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DiscussionOur multisite, long-term study sheds new light on the
nature ofsuccessional dynamics. Our model was able to reproduce
many ofthe general successional trends observed for these
trajectories, yetthe spatiotemporal trajectories in these forest
attributes revealedhigh levels of uncertainty. Even when accounting
for previous landuse and variation in initial conditions at the
first census, the de-terministic and stochastic components of the
model had to besimilar in magnitude to predict successional
trajectories accu-rately. Although variation in successional
pathways has beenwidely acknowledged (14, 15, 18), quantifying the
magnitude ofthis variability has remained an elusive goal. To our
knowledge,our results provide the first quantitatively robust,
multisite as-sessment of the extent of uncertainty during tropical
forest suc-cession using long-term datasets.Previous attempts to
model secondary forest dynamics have fo-
cused on mechanistic approximations based on
species-specificequations to predict changes in species performance
and compo-sition in temperate stands (24, 25). This level of
accuracy is im-practical in tropical forests, where hundreds of
tree species coexist.Here, we intended to represent nature through
models focusing onthe autogenic forces that drive succession. We
acknowledge thatour approach was not strictly mechanistic, yet
parameters esti-mated by the deterministic component of our model
reinforced ourunderstanding of stand dynamics at different stages
of succession.We detected different processes occurring early in
succession, suchas density-dependent thinning and basal area
accumulation as aconsequence of increasing tree recruitment (9).
Likewise, thecompetitive pressure leading to a decrease in species
colonizationrates was also observed in most sites (20). Although
these resultshave already been reported in the literature, they
validate ourmodel and demonstrate that the fitted parameters do
reveal manyof the ecological processes driving successional
dynamics. Ourmodel further reveals previously unexplored patterns,
such as therelationship between species density and rates of change
in stemdensity and basal area. As these associations were only
observed ina few sites, further work may help to disentangle the
mechanismsthrough which species diversity affects biomass dynamics
andthereby ecosystem function (26). Overall, the strength of
theseprocesses differed widely among sites, which may reflect
site-specific characteristics related to species composition, stand
agedistribution, and environmental and landscape factors.Most of
our understanding in tropical successional ecology is
embedded in a deterministic framework where successional
path-ways are primarily driven by autogenic factors, and prior
distur-bances due to anthropogenic land use are typically the only
allogenic,external forces considered (11, 12). Our results showed
that suc-cessional pathways were highly idiosyncratic among nearby
plots ofthe same age since abandonment with similar disturbance
history,and therefore we strongly advise caution in making
inferences aboutrates of vegetation change based on single-time
censuses (27).
Interestingly, the strength of such idiosyncrasy was not linked
to thenature or intensity of prior land use. The sites where
secondaryforests were regenerating after pasture (Brazil 2 and
Costa Rica 1)or shifting cultivation (Mexico wet andMexico dry) did
not show anynotable difference in terms of predictability of
successional trajec-tories compared with sites with forests
regenerating after clear-cut-ting with no subsequent land use
(Brazil 1 and Costa Rica 2). Eventhe forest plots in Nicaragua,
where monitoring in all plots startedsimultaneously soon after the
passage of Hurricane Joan, showedhigh among-plot variability in
their successional trajectories.The complexity of site factors and
their interaction with land use
is widely acknowledged and challenges our ability to predict
suc-cessional pathways at local or regional spatial scales (7).
Topo-graphic variation in soil quality and drainage, distance to
other forestpatches, continuous changes in the surrounding
landscape, initialspecies and functional composition, fire
frequency, and neighbor-hood effects all influence rates of
vegetation change in successionalpathways (27). Moreover, a myriad
of local factors including priorityeffects, invasive species, weed
control, last crop planted, nutrienttreatments, pathogen and
herbivore loads, and persistent edge ef-fects can alter
successional processes and push community trajec-tories in
unpredictable directions (18). Although the nonlinearmodel
developed here does not explicitly include local and land-scape
factors, a key feature of our approach is its high sensitivity
toinitial conditions in stem density, basal area, and species
density,which may account for some of these historical
contingencies. Ourresults underscore the need for future
cross-site, long-term succes-sional studies that consider local,
previously unmeasured factors,and ongoing changes in the
surrounding landscape.By emphasizing the emergent properties of
communities, we
believe that the model developed here represents, to date, the
bestapproach for characterizing tropical forest successional
dynamics.Despite the high levels of uncertainty detected, we might
be able toelucidate the underlying processes behind the patterns
observed,and to anticipate ecosystem change through the further
de-velopment of high-dimensional models (28). Other metrics, such
asspecies and functional dominance could also provide critical
insightsabout the dynamic relation between functional traits and
biomassaccumulation (26). A challenging goal remains to model
multidi-mensional variables such as species or functional
composition. Al-though more complex, successional pathways based on
these metricsmay be more predictable, as species and functional
composition arelikely to be determined by niche-based processes
(29, 30).A potential limitation of our results is that they portray
the first
decades of succession, which reflect the predominant age classes
ofregenerating forests in the Neotropics (12). Only one site,
Mexicodry, comprises secondary stands over 60 y old, and Nicaragua
is thesole site that shows rates of change in forest structure
since thebeginning of the successional process (18). Despite
representingthe most extensive monitoring of forest succession in
the tropics,1015 y of census data are insufficient to capture the
entire range
Mea
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actio
n of
dat
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ints
w
ithin
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0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2
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Fig. 4. Predictability of the stochastic model as afunction of
noise intensity. Mean fraction of the ob-served data points
describing the successional trajec-tories in stem density, basal
area, and species densitythat lie within the envelope generated by
the sto-chastic model at different levels of noise intensity.
8016 | www.pnas.org/cgi/doi/10.1073/pnas.1500403112 Norden et
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over which rates of vegetation change are the most dynamic
(20).Longer time series would allow to evaluate the extent of
conver-gence in successional trajectories within the next decades
(31). Also,they would give essential information about how
secondary forestsrespond to unpredictable climatic events, thereby
elucidating forestresilience and identifying potential tipping
points (32). As abruptchanges are unlikely to be predicted by
deterministic models, ourapproach would provide insights about the
extent of uncertainty inthese atypical cases.Because all natural
systems interact with their surroundings and
are subject to historical contingencies, it is highly
impractical tomeasure all of the factors affecting forest
succession. This com-plexity constrains our capacity to distinguish
ecological signals fromnoise in the successional process. When
patterns do not followdeterministic predictions, ecologists often
invoke stochasticity (3).Such a dual perception of ecological
processes hampers a syntheticunderstanding of community reassembly
in regenerating for-ests (6). Indeed, in complex adaptive systems,
erratic patterns canarise from either stochastic processes that
emerge from seeminglyrandom fluctuations, or from unexplained but
causal variabilityemerging from unknown unknowns (33). Thus,
despite the highlevels of ecological noise observed here, what we
typically view asstochasticity may ultimately be explained by
deterministic factorsthat have not been measured or incorporated.
Our study calls for abetter evaluation of the historical
contingencies and landscapevariables affecting succession. As
regenerating forests have a greatpotential to become important
biodiversity reservoirs and deliverenvironmental cobenefits in an
economically viable manner (34),we urgently need a better
interpretation of research findings re-lated with successional
ecology. If secondary succession is highlycontext dependentas
supported by this studyevaluating theextent of uncertainty in
successional trajectories in relation to local,landscape, and
regional variables will allow a better understandingof the sources
of variation in stand dynamics in human-modifiedlandscapes. New,
integrated approaches that model communitiesas complex systems will
enable prediction of response envelopesto guide the research agenda
and the effective management ofregenerating forests, which
currently encompass more than one-half of all tropical forests
globally (8).
MethodsStudy Sites and Data. We used multitemporal
(repeated-measures) forestdynamics data from multiple lowland
Neotropical terra firme forests locatedin Brazil, Costa Rica,
Mexico, and Nicaragua (more details in SI Methods andTable S1).
Within each site, 415 permanent plots were established in
sec-ondary stands of different ages but with similar disturbance
histories andenvironmental conditions, and were monitored annually
for at least 8 y,except for a few plots that were accidentally
burnt.
Dynamical Modeling. We quantified predictability and uncertainty
duringsuccession in each of the seven sites by using dynamic,
stochastic models to fitthe observed rates of change in stem
density, basal area, and species densitysimultaneously (Fig. 1). We
excluded from this analysis six plots in Brazil 1, onein Brazil 2,
and two in Mexico wet as they were monitored for less than
4consecutive years because of burnings. Stochastic models integrate
pre-dictability (deterministic drivers) and uncertainty
(stochasticity), as described bya system of stochastic differential
equations of the Langevin form (35):
ddt
Xt=gXt, t+hXt, t t, [1]
where X denotes the state of the system at a given time,
characterized by threestate variables: stem density, basal area,
and species density. The right side ofEq. 1 describes the magnitude
of the derivative of each component X over timeas the sum of a
deterministic function gXt, t, and a stochastic functionhXt, t t,
where t stands for terms of Gaussian white noise (Fig.
1).Deterministic component of the model. We first defined the
deterministic com-ponent of Eq. 1, gXt, t, through a system of
first-order differential equa-tions, where the change in each state
variable at time t + depends only on itsstate at time t. For each
site, this system of equations simultaneously models allof the
observed trajectories of stem density, basal area, and species
densityover time, starting at the first census value for each plot.
It must be noted thatinitial conditions for each plot denote the
first observation available for each
trajectory and are not modeled as the beginning of the
successional process att0 after land abandonment. Thus, our results
are not biased by among-plotvariability in stand age at first
census, or by temporal changes in the rates ofchange in forest
structure attributes as succession unfolds.
Because forest structure attributes varied over widely different
ranges, wenormalized these state variables by scaling them between
0 and 1, so thatX = x xmin=xmax xmin. This standardization allowed
values to be adjustedfor different levels of magnitude, without
changing the shape of the distribu-tion. The common z
standardization was not applied to keep the state
variablespositive. Otherwise, this would cause mathematical
artifacts in one of our can-didate models (Eq. 2c), as the system
of equations cannot be solved in the do-main of the real numbers
because a negative number cannot be raised toa fractional
power.
For each of the seven sites, we tested three candidate models.
Linearfunctions are the most commonly used approximation to
investigate therelationships between quantitative variables, and
provide an accurate picturefor assessing local stability points in
the study of dynamical systems. Thus, ourfirst model described a
linear relationship between the rates of change instem density (D),
basal area (BA), and species density (S) as follows:
8>>>>>>>>>>>>>:
dDdt
= a11D+ a12BA+ a13S
dBAdt
= a21D+ a22BA+ a23S
dSdt
= a31D+ a32BA+ a33S
. [2a]
Because this model ignores possible interactions among the state
variables,which are frequent in ecological systems, we formulated a
second, alternativemodel, which included interactions among the
state variables as follows:
8>>>>>>>>>>>>>:
dDdt
= a11D+ a12DBA+ a13D S
dBAdt
= a21BA+ a22BAD+ a23BA S
dSdt
= a31S+ a32SD+ a33SBA
. [2b]
However, most systems are inherently nonlinear in nature.
Indeed, rates ofcommunity change during succession are
characterized by saturating curves(20). Also, as successional
trajectories are highly sensitive to initial condi-tions, small
differences may be amplified and lead to divergent
trajectories,thereby resulting in nonlinearities (23). For these
reasons, the third modelincluded nonlinearities in the system as
follows:
8>>>>>>>>>>>>>:
dDdt
= a11Db11 + a12BAb12 + a13Sb13
dBAdt
= a21Db21 + a22BAb22 + a23Sb23
dSdt
= a31Db31 + a32BAb32 + a33Sb33
. [2c]
The best-fit model was found using a genetic algorithm, a
heuristic method inwhich a randomly created population of
parameters is optimized by means ofcrossover and mutation operators
in a process that mimics natural selection (36).By these means, new
solutions to the system of equations are created, differentfrom the
parent solutions, thereby avoiding local minima. The algorithmwas
run2,000 times until reaching the minimum objective function, i.e.,
the minimumroot-mean-square error of the observations and model
estimates, as follows:
min=Xni=1
jDobs DsimjmaxDobs
+jBAobs BAsimjmaxBAobs
+jSobs SsimjmaxSobs
, [3]
where D, BA, and S are the normalized, temporal trajectories in
stem density,basal area, and species density, respectively, refers
to the objective function,and n, the number of plots within a site.
This method implements a numericalsolution for these first-order
differential equations by defining a t = 0.1 y.This time step is
small enough to assume that the instantaneous rate of changemodeled
can be assimilated to the rate of change at each time step.
We assessed the predictive power of the three candidate models
using theNashSutcliffe model efficiency (NSE) coefficient (37),
defined as follows:
NSE = 1
Xni=1
Yobsi Y
simi
2Xn
i=1
Yobsi Yobs
2, [4]
Norden et al. PNAS | June 30, 2015 | vol. 112 | no. 26 |
8017
ECOLO
GY
-
where Yobsi is the ith observed value, Ysimi is the ith
simulated value, Yobs is
the mean of observed data, and n is the total number of
observations foreach site. The NSE coefficient is a normalized
statistic that determines therelative magnitude of the residual
variance compared with the measureddata variance. This metric
indicates how well the plot of observed versusmodeled data fits the
1:1 line. NSE ranges between and 1, with NSE = 1being the optimal
value. Values between 0 and 1 are generally viewed asacceptable
levels of performance, whereas values 1,the magnitude of the
stochastic component of the model is greater than thatof the
deterministic model. In the stochastic model, each component of X
istreated as a random process. If we set Xti= x+ x, then the
distribution ofthe predicted values of x at the next step of the
trajectory Xti + is given bya Gaussian function with mean x + gx
and SD hx p (35) (Fig. 1). Thecodes for running the stochastic
model were written in Matlab 7.2.
ACKNOWLEDGMENTS. We are grateful to the dozens of field
assistants andcolleagues who participated in the extensive censuses
and assisted with datamanagement (T. V. Bentos, J. and H. Jamangap,
M. Molina, J. Panigua,B. Paniagua, E. Salicetti, E. A. Prez-Garca,
J. Rodrguez-Velzquez, J. Romero,and I. E. Romero-Prez). We thank J.
Chave, R. K. Colwell, A. Duque, S. Levin,A. Ramrez, S. Russo, and
M. Uriarte for insightful comments. E.L.-T. acknowl-edges support
by Panamanian Sistema Nacional de Investigadores, SecretaraNacional
de Ciencia, Tecnologa e Innovacin. The studies were
financiallysupported by National Science Foundation Grants
DEB-1147434, DEB-1147429, DEB-0639393, DEB-9524061, DEB-0135350,
and DEB-0235761;grants from Andrew W. Mellon Foundation and
University of ConnecticutResearch Foundation; Mexican Secretara del
Medio Ambiente y RecursosNaturalesConsejo Nacional de Ciencia y
Tecnologa (CONACYT) 2002-C01-0597, Secretara de Educacin Pblica
(SEP)CONACYT CB-2005-01-51043,CONACYT 2004-168169, and SEP-CONACYT
CB-2009-128136; UniversidadNacional Autnoma de MxicoPrograma de
Apoyo a Proyectos de Investi-gacin e Innovacin Tecnolgica IN216007
and IN213714; and Dutch Nether-lands Organization for Scientific
ResearchNetherlands Foundation for theAdvancement of Tropical
Research W85-326.
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8018 | www.pnas.org/cgi/doi/10.1073/pnas.1500403112 Norden et
al.
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Supporting InformationNorden et al. 10.1073/pnas.1500403112SI
MethodsStudy Sites and Data. In Brazil, study plots were located
about 80km north of Manaus, Amazonas (224S, 5454W). Forests
re-generating after clear-cut with little or no burning were
domi-nated by species of the genus Cecropia, whereas areas
followingpasture were typically dominated by species of the genus
Vismia(31, 39). Because of these differences in land use, we
groupedplots into two sites called Brazil 1 (Cecropia-dominated)
andBrazil 2 (Vismia-dominated), and analyzed them separately.
InCosta Rica, study plots were located in the Sarapiqu
region,Heredia (1025N, 8400W). As in Brazil, secondary forest
plotswere analyzed separately according to their land use, so that
twodifferent sites were defined. Plots established on
abandonedpastures were classified as Costa Rica 1, and those that
wereclear-cut without subsequent use were classified as Costa Rica
2(14). In Mexico, study plots were located in two different
re-gions: one site, adjacent to Montes Azules Biosphere
Reserve,Chiapas (1601N, 9055W), is dominated by wet forests
(Mex-ico wet) (17); the second site, on the Pacific slope of the
Isthmusof Tehuantepec, Oaxaca (1639N, 9500W), is dominated
bydeciduous dry forests (Mexico dry) (16). All Mexican plots
wereestablished on abandoned cornfields. In Nicaragua, study
plotswere located in regenerating primary wet forests of the
Carib-bean coast (123N, 8356W), established after the landfall
ofHurricane Joan of October 1988 (18). The plots comprised byeach
site share similar climate, soil type, land use history,
andvegetation composition.Multiple stemmed individuals were tallied
as a single in-
dividual. We standardized stem density, basal area, and
speciesrichness by plot size, depending upon the site (0.04 ha for
Mexicodry, 0.05 ha for Brazil andMexico wet, 0.1 ha for Nicaragua,
and 1ha for Costa Rica 1 and 1.16 ha for Costa Rica 2). We
measuredspecies richness within each plot as species density
because themodeling procedure already takes into account temporal
changesin stem density, and other commonly used metrics of
diversity
such as rarefied richness, which controls for differences in
stemdensity, would have been redundant. Although we acknowledgethat
the speciesarea relationship is not linear, this standardi-zation
was only performed in Brazil 1 and Brazil 2, which hadvariable plot
sizes. In these two sites, as the plots were very small(0.010.06
ha), the relationship between species richness andarea was linear
(39). We chose the smallest plot size to performthe standardization
to avoid improper extrapolation of speciescounts from smaller to
larger areas. In Costa Rica 2, all plotswere 1.16 ha, except for
one, which increased from 0.33 to 1.16ha in the third census
period. For this site, we omitted speciesdensity values in the
first two censuses for the smallest plot.
Effects of Stand Age and Plot Factors. For each site, we used
one-way random-effects ANOVA for repeated measures to test forthe
effect of stand age since abandonment (fixed effect) and
plotidentity (random effect) on stem density, basal area, and
speciesdensity in each site. These analyses were performed using
the Rstatistical package (40).
Importance of Plot Size and of Time Series Length. We assessed
theeffect of plot size and of time series length in the level of
noiseneeded to increase model predictability. To assess the effect
ofplot size, we subsampled the plots from Costa Rica 1, one of
thesites with the biggest plots, and ran the stochastic model
insubplots of 500, 1,000, 2,000, 5,000, and 10,000 m2. We
thenevaluated whether the fraction of data points describing
thetrajectories in stem density, basal area, and species density
thatwere within the envelope generated by the stochastic model for=
0.5 and = 1 varied depending upon sample size.We assessed the
effect of time series length by subsampling
either the first or the last 5 y of data for each plot within
each site.We then compared the outcome of the stochastic model
based onthese two data subsets and the one using the original data
for= 0.5 and = 1.
Norden et al. www.pnas.org/cgi/content/short/1500403112 1 of
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Age
Stem density Basal area Species density
0
0.2
0.8
0 10 20 30
0 10 20
0 10 20 30 40 50
0 10 20 30 0 10 20 30
0 10 20 0 10 20
0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 0 10 20 30 0 10 20 30
0 20 40 60 0 20 40 60 0 20 40 60
0 5 10 15 20 0 5101505 025101 20
Nic
arag
uaM
exic
o w
etM
exic
o dr
yCo
sta
Rica
2Co
sta
Rica
12lizarB
1lizarB
0.4
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1
0
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1
0
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0
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0.6
1
Fig. S1. Modeled successional pathways of community attributes.
For each plot within each site, temporal changes in the
standardized values in stem density,basal area, and species density
are plotted against age since abandonment (thin lines connecting
blank dots). Continuous thick curves show the fitted de-terministic
model (Eq. 2c). The gray envelopes are defined by the 1,000
trajectories generated by the stochastic model for = 0.5.
Norden et al. www.pnas.org/cgi/content/short/1500403112 2 of
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500 1,000 2,000 5,000 10,000 500 1,000 2,000 5,000 10,0000
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
=0.5 = 1
Mea
n fr
actio
n of
dat
a po
ints
w
ithin
the
mod
el e
nvel
ope
stem densitybasal areaspecies density
Plot size (m )2
Fig. S2. Effect of plot size on stochastic model performance.
Mean fraction of the observed data points describing the
successional trajectories of stemdensity, basal area, and species
density that are within the model envelopes generated by the
stochastic model, depending upon plot size, for =0.5 and = 1.Plot
sizes represent a subsampling of the 1-ha plots from Costa Rica 1.
Plot size does not have a systematic effect on the predictability
(fraction of data pointswithin the model envelope) of any of the
forest attributes evaluated.
0
0.2
0.4
0.6
0.8
1
0
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1
Brazil
1
Brazil
2
Costa
Rica
1
Costa
Rica
2
Mexic
o wet
Mexic
o dry
Nicara
gua
Brazil
1
Brazil
2
Costa
Rica
1
Costa
Rica
2
Mexic
o wet
Mexic
o dry
Nicara
gua
=0.5 = 1
full time seriesfirst 5 yearslast 5 years
Mea
n fr
actio
n of
dat
a po
ints
with
in th
e m
odel
env
elop
e
Stem density
Basal area
Species density
Fig. S3. Effect of time series length on stochastic model
performance. For each site, mean fraction of the observed data
points describing the successionaltrajectories of stem density,
basal area, and species density that are within the model envelopes
generated by the stochastic model, depending upon timeseries
length, for = 0.5 and = 1. If time series length had an effect on
model performance, shorter times series would have been expected to
show lowerpredictability, that is, a smaller fraction of observed
data points would occur within the model envelopes. Time series
length does not have a systematic effecton the predictability
(fraction of data points within the model envelope) of any of the
forest attributes evaluated, at any of the sites.
Norden et al. www.pnas.org/cgi/content/short/1500403112 3 of
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Table S1. Description of each site
SiteNo. ofplots Plot area, ha
Stand age atfirst census, y
Measurementperiod
Total no. of treespecies recorded Previous land use
Minimumdbh, cm
Brazil 1 (31, 39) 15 0.01250.06 219 19992010 398 None 3Brazil 2
(31, 39) 13 0.010.06 211 19992010 246 Pasture 3Costa Rica 1 (14) 6
1 1235 19972012 366 Pasture 5Costa Rica 2 (14) 4 0.331.16 125
19872011 360 None 10Mexico wet (17) 10 0.05 217 20002010 206 Fallow
cornfields 5Mexico dry (16) 12 0.04 360 20032012 75 Fallow
cornfields 5Nicaragua (18) 12 0.1 2 19902007 328 None 5
Characteristics of the study plots for each of the seven sites.
Literature references are in parentheses next to the site.
Table S2. Relative effects of plot and stand age on the forest
structure attributes
Forest attribute Term Brazil 1 Brazil 2 Costa Rica 1 Costa Rica
2 Mexico wet Mexico dry Nicaragua
Stem density Plot 87.6 69.8 86.8 29.6 60.1 87.2 26.3Age 0.3*
0.8** 0.3** 1.0ns 1.7** 4.1*** 30.2***
Plot Age 7.1*** 10.7*** 11.1*** 7.5ns 15.4** 7.8*** 18.0***Basal
area Plot 87.6 78.8 86.2 77.7 53.6 86.6 60.1
Age 103 ns 4.7*** 10.8*** 14.5*** 17.0*** 10.1*** 31.7***Plot
Age 7.8*** 11.9*** 1.6*** 4.4*** 11.3*** 2.1*** 3.8***
Species density Plot 93.9 73.4 95.1 68.1 45.3 95.1 66.6Age 103
ns 7.8*** 0.4*** 30.6*** 16.3*** 2.6*** 14.1***
Plot Age 2.5*** 5.9*** 3.5*** 0.8*** 11.5*** 1.2*** 8.4***
Results from one-way repeated-measures ANOVA testing, for each
site, the effect of plot identity (random effect) and stand
age(fixed effect) on stem density, basal area, and species density.
Reported are the percentage of variance explained by age, plot
factors,and their interaction. Plot factors were not tested for
significance because they were treated as random effects (***P <
0.001; **P >>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
18 AIC 983.2 919.5 602.1 264.6 626.8 1,233.6 1,020.3NSE 0.98
0.92 0.95 0.92 0.89 0.98 0.93
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
16 AIC 1,077.7 1,004. 2 615.7 288.5 608.7 1,154.1 964.4NSE 0.98
0.93 0.93 0.94 0.88 0.97 0.94
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
16 AIC 1,038.7 963.4 627.3 299.7 569.4 1,260.8 852.8NSE 0.98
0.93 0.93 0.94 0.88 0.98 0.94
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
16 AIC 1,037.8 1,008.2 570.6 266.1 614.2 1,208.5 848.1NSE 0.98
0.94 0.95 0.95 0.88 0.98 0.90
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
14 AIC 1,001.8 978.8 594.8 286.7 627.5 1,384.9 964.7NSE 0.98
0.94 0.95 0.96 0.89 0.98 0.94
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
14 AIC 1,050.5 1,118.9 576.6 294.9 640.3 1,332.8 1,096.8NSE 0.98
0.94 0.95 0.96 0.88 0.98 0.94
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
14 AIC 1,108.1 853.7 646.3 305.3 604.1 1,421.9 861.8NSE 0.98
0.92 0.95 0.94 0.89 0.98 0.89
8>>>>>>>>>>>>>:
dDdt
= fD,BA, SdBAdt
= fD,BA, SdSdt= fD,BA, S
12 AIC 1,007.4 926.1 686.9 310.4 621.1 1,389.1 820.8NSE 0.98
0.94 0.95 0.95 0.89 0.97 0.93
These were (i) the effect of species density (S) on the rates of
change in stem density (D), (ii) the effect of species density (S)
on the rates of change in basalarea (BA), and (iii) the effect of
basal area (BA) on the rates of change in species density (S).
Eliminated terms are in bold in the equations. We compared
theoriginal nonlinear model with seven models resulting from the
elimination of each of these three terms separately, and all its
possible combinations. For eachcandidate model, we report its
number of parameters, the NashSutcliffe efficiency coefficient
(NSE), and the approximate Akaike information criterion
(AIC).Values in bold indicate the best-fit model for each site.
Norden et al. www.pnas.org/cgi/content/short/1500403112 5 of
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Table
S5.
Parameter
estimates
ofthebest-fitmodel
based
onthegen
eticalgorithm
optimizationmethod
Parameter
estimate
Brazil1
Brazil2
CostaRica1
CostaRica2
Mexicowet
Mexicodry
Nicarag
ua
a 11
1.42
(0.89
0.03
)0.19
(0.86
0.02
)0.23
(0.82
0.02
)+0.02
(0.71
0.02
)1.38
(0.72
0.02
)0.08
(0.87
0.02
)0.74
(0.76
)a 1
2+0.10
(+0.06
0.03
)+0.07
(+0.29
0.01
)+0.28
(+0.39
0.02
)0.16
(+0.20
0.02
)+0.92
(+0.32
0.02
)+0.02
(+0.24
0.02
)+0.23
(+0.40
0.02
)a 1
3+0.27
(+0.01
0.02
)0
00
0+0.04
(+0.22
0.02
)0
a 21
+0.03
(+0.09
0.02
)+0.04
(+0.24
0.02
)+0.13
(+0.38
0.02
)+0.27
(+0.46
0.02
)+1.04
(+0.26
0.02
)0.06
(+0.30
0.02
)+0.21
(+0.17
0.02
)a 2
20.34
(0.63
0.02
)0.09
(0.83
0.02
)0.13
(0.59
0.02
)0.16
(0.59
0.02
)0.71
(0.80
0.02
)+0.04
(0.41
0.02
)1.38
(0.76
0.02
)a 2
30
+0.06
(+0.34
0.02
)0
00.47
(+0.28
0.02
)0
+0.02
(+0.26
0.01
)a 3
1+0.47
(+0.25
0.03
)+0.04
(+0.32
0.02
)+0.07
(+0.19
0.02
)+0.05
(+0.30
0.01
)+0.18
(+0.38
0.01
)+0.21
(+0.30
0.01
)0.47
(+0.34
0.01
)a 3
20
00
00
00
a 33
+0.04
(0.46
0.02
)0.09
(0.56
0.02
)0.03
(0.46
0.02
)+0.02
(0.53
0.02
)0.88
(0.64
0.02
)0.16
(0.48
0.02
)+0.16
(0.70
0.02
)b11
+2.95
(+0.73
0.03
)+0.69
(+1.19
0.02
)+1.90
(+1.32
0.03
)+0.16
(+1.32
0.03
)+1.64
(+1.19
0.02
)+1.16
(+1.19
0.02
)+2.57
(+1.32
0.02
)b12
+0.35
(+0.81
0.03
)0.48
(+0.78
0.02
)+3.05
(+1.29
0.03
)+6.24
(+1.28
0.03
)+0.67
(+0.75
0.02
)+0.07
(+1.08
0.02
)+0.10
(+0.71
0.02
)b13
+2.29
(+0.78
0.03
)0
00
0+0.36
(+1.01
0.02
)0
b21
+1.62
(+0.85
0.03
)2.11
(1.00.02
)+0.70
(+1.13
0.03
)+2.14
(+1.29
0.03
)+0.26
(+1.04
0.02
)+4.72
(+1.15
0.02
)+1.54
(+1.07
0.01
)b22
+5.47
(+0.86
0.03
)+0.48
(+1.07
0.02
)+1.67
(1.31
0.03
)+2.58
(+1.55
0.03
)+0.66
(+1.16
0.02
)+0.33
+1.24
0.02
)+4.11
(+0.92
0.02
)b23
00.27
(+0.91
0.02
)0
0+0.18
(+0.91
0.02
)0
+0.29
(+0.91
0.02
)b31
+4.10
(+0.69
0.03
)+0.13
(+1.02
0.02
)+6.3(+1.24
0.03
)+0.53
(+1.14
0.03
)+0.22
(+0.88
0.02
)+1.96
(+1.28
0.02
)+2.24
(+0.89
0.01
)b32
00
00
00
0b33
+0.48
(+0.81
0.04
)+1.81
(+1.25
0.02
)+3.11
(1.24
0.04
)+4.49
(+1.86
0.04
)+3.24
(+1.22
0.03
)+2.33
(+1.23
0.03
)+0.04
(+1.29
0.01
)
Thegen
etic
algorithm
isthebestoptimizationmethodforestimatingnonlin
eardyn
amic
operators,forwhichan
analytic
optimizationisnotpossible.Becau
sethismethoddoes
notprovideconfiden
ceintervalsfortheparam
eter
estimates,also
reported
arethemea
nan
dtheSE
oftheparam
eter
values
ofthe10
0(5%)bestruns.
Norden et al. www.pnas.org/cgi/content/short/1500403112 6 of
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Table S6. Relationship between the observed and predicted values
of stem density, basal area, and species density within each
site
SiteForest structure
attribute Range in the slopes Significantly positive
Significantly negative Nonsignificant
Brazil 1 (8) Stem density 0.16 to +0.44 0 0 8Basal area 0.47 to
+0.61 0 0 8Species density 0.21 to +0.82 3 0 5
Brazil 2 (12) Stem density 2.40 to +4.33 11 1 0Basal area +0.03
to +11.78 6 0 6Species density 0.02 to +1.10 3 2 7
Costa Rica 1 (6) Stem density 0 to +0.67 1 0 5Basal area +0.01
to +0.99 3 0 3Species density 0.10 to +0.07 0 0 6
Costa Rica 2 (4) Stem density +0.07 to +0.22 1 0 3Basal area
+0.02 to +0.81 2 0 2Species density +0.32 to +0.98 1 0 3
Mexico wet (13) Stem density 0.35 to +1.81 3 2 8Basal area 0 to
+0.97 6 0 7Species density 0.87 to +1.69 1 0 12
Mexico dry (12) Stem density 0.44 to +1.24 2 1 9Basal area +0.07
to +1.01 5 0 7Species density 0.01 to +1.28 5 0 7
Nicaragua (12) Stem density 0.33 to +0.74 0 0 12Basal area 0 to
+1.43 5 0 7Species density 0.04 to +0.76 0 0 12
Total no. of cases 58 6 137
The results summarize a linear ANCOVA model relating the
predicted to the observed values in stem density, basal area and
species density in each site,based on the best-fit nonlinear model
(Eq. 2c), and including plot as a covariate. Reported are the range
in the estimated slopes relating observed to predictedvalues for
each plot, the number of slopes that were significantly positive,
significantly negative, and nonsignificantly different from zero.
The numbers inparentheses refer to the number of plots included in
this analysis for each site.
Norden et al. www.pnas.org/cgi/content/short/1500403112 7 of
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