journa l homepage: www.e lsev ier .com/ locate /eco lmodel
pplication of a spatial meta-population model with stochastic parameters to theanagement of the invasive grass Nassella trichotoma in North Canterbury,ew Zealand
lex Jamesa, Richard Browna, Britta Basseb,∗, Graeme W. Bourdôtb, Shona L. Lamoureauxb,ick Robertsc, David J. Savilled
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New ZealandAgResearch Limited, Lincoln, Private Bag 4749, Christchurch 8140, New ZealandCentre for Mathematical Biology, Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag 102904, North Shore Mail Centre, Auckland,ew ZealandSaville Statistical Consulting Limited, P.O. Box 69192, Lincoln 7640, New Zealand
r t i c l e i n f o
rticle history:eceived 11 August 2010eceived in revised form3 November 2010ccepted 28 November 2010vailable online 22 December 2010
eywords:ispersalathematical modelassella tussockopulation growtherrated tussockeed control
a b s t r a c t
Optimising the management of invasive plants requires the identification of the population size outcomesfor alternative management strategies. Mathematical models can be useful tools for making such man-agement strategy comparisons. In this paper we develop a generic landscape meta-population model andapply it to the weedy grass, Nassella trichotoma, an invasive species occupying approximately 800 landparcels, predominantly pastoral farms, in the Hurunui district, North Canterbury, New Zealand. Empiri-cal evidence reveals that this meta-population is currently stable (at a median density of 6 plants ha−1)under a community strategy requiring manual removal (termed ‘grubbing’) of plants annually from allland parcels. Reduction in population size requires an alternative management strategy. Field data, col-lected over a 12 year period, were used to provide stochastic parameter values for land parcel size,carrying capacity, rates of local population growth and grubbing.
The model reveals that at steady state, the most significant contribution to population growth on a landparcel comes from within the land parcel itself; the expected annual per capita growth on an individualland parcel is 4 orders of magnitude greater than the expected annual contribution from plants arisingfrom other land parcels. This result implies that many of the farms currently supporting N. trichotomamay pose little or no threat to, nor are threatened themselves by, other farms infested by the weed.
However, the steady state distribution (of the weed’s population density) was sensitive to the spread rate,revealing a need for data on this process. It was also sensitive to how any increase in the grubbing rate isdistributed; increasing it via a uniform distribution U(0, 1) where all rates between 0 and 100% year−1 areequally probable did not affect the steady state, whereas increasing the rates via the uniform distributionU(0.25, 0.75) resulted in fewer farms with high population densities. In general the model provides a
ects o
basis for exploring the effN. trichotoma.
. Introduction
Mathematical models of weed population dynamics and inva-ion are useful tools for (i) identifying population parameters withhigh impact on future population distributions, (ii) estimating
nknown parameters and (iii) for comparison of managementtrategies (Cousens and Mortimer, 1995; Hester et al., 2010;ortimer et al., 1989). In this paper we develop a generically
pplicable meta-population model for the spread of an invasive
plant between land parcels (i.e. farms and other land managementunits) using logistic (bounded) population growth equations sim-ilar to those discussed in detail by Hanskii (1998) and Caswell(2001). The model is applied (Fig. 1) to the local scale demograph-ics and spread of Nassella trichotoma (Nees) Arechav. (commonlycalled nassella tussock or serrated tussock) in the Hurunui dis-trict (Fig. 2), of North Canterbury in the South Island of NewZealand. Infestations are present across the entire region with
occurrence patterns described by Kriticos et al. (2004). Our over-all aim is to show how the combination of theory and data can beused to understand population growth and spread and to developa realistic and hence useful management tool for an invasiveplant.
ig. 1. Conceptual diagram of the model in Eq. (1) describing the underlying demo-raphic processes driving the dynamics of the N. trichotoma meta-population andhe relationship to management decision making.
N. trichotoma is a perennial grass of South American origin thatas invaded drought-prone pastures in South Africa, Australia andew Zealand. It is non-palatable to sheep and other livestock andan cause significant economic loss to the pastoral industry. Sincets arrival in New Zealand over 100 years ago, it has establishedtself as a serious pastoral weed. Historically it achieved such highensities (approximately 35,000 plants ha−1) in the Canterburynd Marlborough regions that pastoral farming was compromisedHealy, 1945). The Nassella Tussock Act, passed in 1946, legislatedor a central government-funded national eradication programme.he weed was managed nationally until 1990 from which time it
as been managed under Regional Pest Management Strategiesrovided for by the New Zealand Biosecurity Act (Anonymous,993). Control is effected by grubbing, the manual removal of plantsy digging them out with a grubber. This labour-intensive control
ig. 2. Map of the study area: the Hurunui District of the North Canterbury Regionf New Zealand.
ng 222 (2011) 1030–1037 1031
is undertaken by land managers, or their contractors, with compli-ance and education the jurisdiction of regional councils. Althoughdensities are now considerably lower than historically, eradicationhas not been achieved and, judging from an analysis of historicalgrubbing records from 1966 until 1988 showing that populationsin North Canterbury had stabilised over this period, eradicationis an unrealistic goal (Bourdôt et al., 1992). This latter conclusionis further supported by ten years of population monitoring datafrom the Hurunui district indicating that the average plant densityhas remained steady for the last decade under an annual grubbingregime (Bourdôt and Saville, 2007).
A key question for regional decision makers is: what is theeconomically optimal grubbing strategy? In an earlier attempt toanswer this question, Denne (1988) developed a bio-economicmodel that gave maximum net return to New Zealand when thefrequency of grubbing was reduced from once every year to oncein three or four years. However, the practical and social difficultiesof such a strategy, untested assumptions underlying the popula-tion dynamics sub-model and the concern by decision-makers thatN. trichotoma populations might regress back to their historicalhigh densities under such a reduced grubbing effort, precludedits implementation. This concern was heightened by the changefrom nationally funded control to regional programmes requiringrate-payer approval. In order to answer the question above, a bet-ter understanding of the population dynamics of N. trichotoma wasneeded. New research to this end was begun in 1998 to quantify (a)local population growth rates and the underlying demographic pro-cesses (Lamoureaux et al., 2006), (b) grubbing-induced mortalityrates (Verkaaik et al., 2006) and (c) current land parcel infestationsizes (see Bourdôt and Saville, 2005).
We use these data, gathered over a 12 year period, to parame-terise the logistic meta-population model and explore mechanismsof local (within a land parcel) population dynamics and non-local(between land parcels) spread in N. trichotoma. The infested land inthe Hurunui district is divided into approximately 800 land parcelseach managed separately by the land owner or lease holder. In themodel, each land parcel is considered to be one sub-populationof a meta-population, each with an associated number of N. tri-chotoma plants. The model provides conservation equations for thenumber of plants on each land parcel changing over time due tolocal per capita growth and per capita spread between land parcels.While there are a number of seed dispersal mechanisms responsi-ble for the establishment of N. trichotoma offspring plants on oneland parcel originating from a parent plant on another land parcel(wind, stock, feed, machinery, etc.), the model implicitly accountsfor these in a single Gaussian dispersal kernel dependent on dis-tance between land parcel centres. We consider two classificationsof mortality, grubbing induced mortality (the manual removal ofplants by humans) and natural mortality. The former is accountedfor explicitly in the model because it has been measured experi-mentally (Verkaaik et al., 2006), is the dominant form of mortalityand is a control parameter. The latter is known to exist becausemonitored uncontrolled populations of N. trichotoma sporadicallydeclined from one year to the next (Lamoureaux et al., unpublisheddata). We account for this in the model by allowing annual growthrates to be negative.
2. Materials and methods
2.1. Generic meta-population model and theory
Suppose we have a region containing M land parcels. Let xi(t) bethe total number of plants on land parcel i at year t. The growth andspread of the plant populations give rise to a dynamically chang-ing meta-population that is modelled by the ordinary differential
1032 A. James et al. / Ecological Modelling 222 (2011) 1030–1037
Table 1Summary of model parameters (Eq. (1)).
Parameter Definition Units Distribution Parameter values
t Time years –M Number of land parcels – – 100i, j Land parcel indices – – –xi Total number of plants on land parcel i Plants per land parcel Ci/2 (when t = 0) –x∗
iSteady state total number of plants on land parcel i Plants per land parcel – –
ri Local land parcel i per capita growth rate year−1 Log-normal r = 0.1 year−1, variance 0.05 year−2
r−1 −4 −1 2
r−1
ts pe
e
wcappii1acmi
wadl
ieaslpr
tp
2
wtpakrcteoaw
We now apply the generic heterogeneous model to N. trichotomawith removal due to grubbing. Fig. 3 shows the current distributionof the per land parcel density of N. trichotoma (plants ha−1) for the789 land parcels in the Hurunui district of North Canterbury, NewZealand, estimated by transforming data on plants grubbed per land
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rel
ativ
e fr
eque
ncy
kij Spread rate (between land parcels i and j) yeagi Local land parcel i per capita grubbing rate yeaCi Carrying capacity Plandij Distance between land parcels i and j centres km
quation
dxi
dt=
⎛⎜⎜⎜⎜⎝rixi +
M∑j = 1j /= i
kijxj
⎞⎟⎟⎟⎟⎠
(1 − xi
Ci
)− gixi, (1)
here, on land parcel i, ri is annual local per capita growth, Ci is thearrying capacity (measured as the total number of plants), gi is thennual per capita removal of plants (death rate) and kij is the annualer capita spread rate of plants arriving in parcel i via seeds fromarents in parcel j. The source terms are subject to a logistic carry-
ng capacity and this assumes that plants are less likely to recruitnto a population as its density increases (Cousens and Mortimer,995; Harper, 1977). If an external source exists it is treated simplys another land parcel, most likely with a zero removal rate. Theonceptual model and its relationship to management decision-aking, through variation of the removal parameter, are illustrated
n Fig. 1.Here the local population growth rate can be zero or negative,
hile per capita dispersal is strictly non-negative. Thus the modelllows the number of plants on a given land parcel to be entirelyriven by other populations if the per capita growth rate on that
and parcel is not positive.The model also assumes that the population on each land parcel
s homogeneous and that the differences occur between differ-nt land parcels. If many land parcels are being considered thisssumption is reasonable because each land parcel covers only amall fraction of the total area and therefore the variation betweenand parcels will dominate. There is no explicit size structure of thelants in the population and so local per capita growth and removalates apply equally to all plants.
Thus the modelled population growth is dependent only onhe total number of plants. The nomenclature in Table 1 providesarameter descriptions and details of units.
.2. Identical land parcels (a homogeneous region)
A simple, though unrealistic, analytical case of the model ishere the region is homogeneous and all land parcels are iden-
ical, i.e. they have identical carrying capacities and identical localer capita growth rates (assumed positive here), removal (death)nd spread rates. Thus the rate of spread between land parcels isij = k plants year−1 when i /= j, the local per capita growth ratei = r year−1, the carrying capacity Ci = C plants and the local perapita removal rate is gi = g year−1. The asymptotic behaviour of
he meta-population distribution can be found by examining theigenvalues of the Jacobian matrix evaluated at zero for the systemf differential equations given in Eq. (1) where i = 1, 2, . . ., M. Therere two distinct eigenvalues �1 = r + (M − 1)k − g and �2 = r − k − g,ith �2 repeated M − 1 times. The larger eigenvalue, �1 passes
r land parcel Exponential C = 430 × 35,000Gamma � (� = 3, ˇ = 1)
through zero at g* = r + (M − 1)k. If the removal rate falls below g*the population on each land parcel will grow to the steady state
x∗ = C(r + k(M − 1) − g)r + k(M − 1)
. (2)
If the removal rate is above g* on each land parcel then the popu-lation on each land parcel will eventually decline to zero.
2.3. Non-identical land parcels (a heterogeneous region)
The more realistic case of non-identical land parcels, i.e. theregion is heterogeneous, is a little more complex. However, numer-ical explorations have shown that the zero solution still exists and,provided all the individual removal rates satisfy the criteria in Eq.(3), it is the only stable fixed point.
gi > g∗i = ri +
M∑j = 1j /= i
kij. (3)
As individual removal rates fall and the criteria are not satisfied onevery land parcel at some point the non-zero solution becomes pos-itive. A sketch proof confirming this is available from the authors.
2.4. Application
0 50 100 150 200 250 >3000
Density (plants ha−1)
Fig. 3. Frequency plot showing the estimated density (plants ha−1) of N. trichotomaon each of 789 land parcels in the Hurunui district of New Zealand.
odelli
paalowTisvcitvd
tsmlir(rs
eeaptnDostd“WAtd
blisIpd
2
otasbou2
2
l
A. James et al. / Ecological M
arcel in the spring of 2002 (supplied by Environment Canterburynd described in Bourdôt and Saville, 2007). There are large vari-tions in density (plants ha−1) between land parcels; over half theand parcels have less than 10 plants ha−1 and a significant numberf land parcels have over 100 plants ha−1 with 3 land parcels (allith area less than 20 ha) having densities of over 1000 plants ha−1.
he mean N. trichotoma density ((total plant numbers)/(total area))s approximately 20 plants ha−1. The mean density calculated as theimple (unweighted) average of the individual land parcel densityalues, is higher at approximately 31 plants ha−1. The median N. tri-hotoma density per land parcel is 5.8 plants ha−1. A small changen the number of high density land parcels can significantly changehe mean value, making it more appropriate to consider the medianalue rather than the mean as it is less affected by the tail of theistribution.
From an analysis of historical grubbing records it is evident thathe density of N. trichotoma plants in the Hurunui district declinedteadily during the 1960s and 1970s (Bourdôt et al., 1992). This isost likely a reflection of the Nassella Tussock Act, passed in Par-
iament in 1946, that made control mandatory for all land ownersn the area. Around 1980 the average density stabilised and hasemained at these or lower levels through until the present dayBourdôt and Saville, 2007). This allows us to presume that cur-ently the N. trichotoma density distribution has reached a steadytate.
In order to capture the heterogeneity between land parcels thexample presented in Section 2.1 is extended to allow the param-ter values to vary between land parcels, i.e. each land parcel isssigned its own local per capita growth rate, grubbing rate, landarcel size etc. The model’s parameters are random variables andheir values were taken from specified distributions because it wasot feasible to measure them for every land parcel in the region.ata, where available, were used to find likely distributions for eachf the parameters. For every land parcel, the parameters were firstampled from their chosen distribution. Then with this parame-er set the predicted steady state distribution was found. This wasone by setting the right hand side of Eq. (1) to zero and using atrust-region dogleg” method supplied via the MATLAB (The Math-
orks Inc.) function fsolve to solve the set of non-linear equations.starting density of half carrying capacity was used. These simula-
ion runs were then repeated to give the model’s steady state plantensity distribution across the land parcels.
For some of the model’s parameters, the values and their distri-utions have been reliably estimated, for example land parcel size,
ocal per capita growth rates and grubbing rate. Carrying capac-ty is less well quantified and other parameters, specifically thepread of N. trichotoma between land parcels remain unknown.n this case mechanisms leading to different distributions of thearameter were explored and the effect of this on the predictedistribution of N. trichotoma numbers was examined.
.5. Model inputs
For each model parameter, where data were available, a numberf possible candidate distributions were chosen from a list of dis-ributions (exponential, normal, log-normal, uniform, power lawnd gamma). Selection here was based simply on the distributionhape of the data. The best fit parameters for each candidate distri-ution were then determined formally using log-likelihoods. Anverall best fitting distribution was chosen from the candidatessing Akaike information criterion (AIC) (Burnham and Anderson,
002).
.5.1. Carrying capacityAs xi is the total number of plants, the carrying capacity of each
and parcel is equal to the carrying density (plants ha−1) multiplied
ng 222 (2011) 1030–1037 1033
by the land parcel size (ha). The mean land parcel size is 430 ha andthe best fit distribution is an exponential with mean 430. The car-rying density for N. trichotoma is approximately 35,000 plants ha−1
(i.e. 3.5 plants m−2) (Healy, 1945; Taylor, 1987a). Combining thiswith the land parcel size distribution gives a distribution for thecarrying capacity (plants per land parcel) of N. trichotoma
Ci(s)∼ 1
Cexp
(− s
C
), s ∈R+ (4)
where the mean number of plants a land parcel can support isC = 430 × 35,000. Fig. 4a shows the distribution of land parcel sizesfor the Hurunui data set with the best fit distribution used in thesimulations.
2.5.2. Annual per capita growth ratesThe parameter ri encapsulates all new plants on the land parcel
arising from a parent plant on that land parcel. In the absence ofdispersion and grubbing but with the possibility of natural mortal-ity ri can be interpreted as the annual per capita local growth ratei.e.
ri = (property i plants in year t) − (property i plants in year t − 1)property i plants in year t − 1
. (5)
A distribution of ri, calculated via Eq. (5), was obtained using twiceyearly counts of N. trichotoma plants within ten netted (i.e. pre-venting dispersion and grubbing) 3 m × 3 m plots in the Hurunuiregion over a period of seven years (Lamoureaux et al., unpublisheddata). Here we have assumed that this distribution of growth ratesis valid at the land parcel scale. Fig. 4b shows the distribution ofannual per capita local growth and the best fit log-normal distri-bution used in the simulations. The probability density functionof the three-parameter log-normal distribution is given in Eq. (6)where � and � are the mean and standard deviation of the corre-sponding normal distribution and � is the shift parameter (Cohenand Whitten, 1980). The best fit log-normal has parameter val-ues � = −0.70 year−1, � = 0.41 year−1 and � = −0.42 year−1 givinga log-normal distribution mean of r = 0.1 year−1 and variance of0.05 year−2.
ri(s)∼ 1
(s − �)�√
2�exp
(− (log(s − �) − �)2
2�2
), s ∈R+ (6)
2.5.3. Grubbing ratesVerkaaik et al. (2006) estimated grubbing rates based on counts
of plants on transect lines before and after grubbing on 24 landparcels in the Hurunui district. The mean grubbing rate from thedata was 0.34 year−1. The best fit distribution in this case is a uni-form distribution with mean 0.43 year−1 where all values between0 and 0.86 are equally possible, hence
gi∼U(0, 0.86). (7)
The data distribution and the best fit distribution used in the sim-ulations are shown in Fig. 4c.
2.5.4. Spread ratesN. trichotoma spreads from one land parcel to another as seeds
via a variety of mechanisms including farm machinery and stock.However, the main mechanism is the wind-dispersal of seed-bearing panicles. Anecdotal accounts tell of large numbers ofpanicles floating through the air when plant densities were highduring the early/mid 1900s. However these accounts have never
been verified. To our knowledge the only scientific study that hasbeen undertaken on wind dispersal in N. trichotoma simply mea-sured the distance travelled by panicles in a 10 min interval (Taylor,1987b). So there is very little data on the dispersal of the paniclesof N. trichotoma.
1034 A. James et al. / Ecological Modelling 222 (2011) 1030–1037
0 500 1000 1500 >20000
1
2
3
4
5
6
7
x 10−3
Property size (ha)
Rel
ativ
e fr
eque
ncy
aDataBest fit exponential
0 1 2 30
0.5
1
1.5
2
Annual local per capita growth ri
bDataBest fit log−normal
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Annual local per capita grubbing gi
Rel
ativ
e fr
eque
ncy
cDataBest fit uniform
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
0.025
Distance (km)
dDataBest fit gamma
F in theg rubbinc ven in
appnlcdtpatsfsdob2pga
k
pfpdn
ig. 4. Frequency distributions of data from 789 N. trichotoma-infested land parcelsrowth rates (year−1) from sample field sites; (c) local (per land parcel) per capita gentres. Details of the corresponding best fit distributions (curves; line in (c)) are gi
The model requires a distribution of values for the per capitannual spread rates kij, parameterising the number of offspringlants establishing in land parcel i from a parent plant on landarcel j. A simple starting point is to assume that the arrival ofew plants on one land parcel is via the spread of seed from other
and parcels and that this is related to the distance between theentres and the sizes of the land parcels. If two land parcels are aistance dij (km) apart then, assuming a Gaussian dispersal kernel,he amount of seed spread from land parcel i to land parcel j isroportional to exp(−dij
2/˛) for some constant ˛ (km2). It is thenssumed that the number of new plants is directly proportional tohe number of seeds that arrived there. In addition to the Gaussianpread a large land parcel may be expected to receive more seedrom neighbouring land parcels than a small one. So the disper-al kernel is also multiplied by the relative land parcel size Ci/C. Aistribution for dij (km) can be found by analysing the distributionf distances between land parcel centres. The best fitting distri-ution, using the methods described at the beginning of Section.5 was found to be a gamma distribution � with integer shapearameter � = 3 and inverse scale parameter ˇ = 1 (Fig. 4d). Thisives a distribution for the spread of plants from one land parcel tonother
ij∼Ci
Cq exp
(−
d2ij
˛
)where dij(s)∼� (s, � = 3, ˇ = 1), s ∈R+ (8)
Here Ci is distributed as Eq. (4) and ˛ (km2) parameterises the
roximity of neighbours that will contribute most to the spread, i.e.or small ˛ most new arrivals will have originated from nearby landarcels. Parameter q (year−1) is the total number of new arrivalsue to spread between land parcels (N.B. q does not equal the meanumber of plants due to spread between land parcels). Values for
Hurunui district for: (a) land parcel sizes (ha); (b) local (per land parcel) per capitag rates (year−1) for 24 sample land parcels; (d) distances (km) between land parcelTable 1.
the parameters ˛ and q are unknown but these are estimated bycomparing model and data (see Section 3.1).
Fig. 5a shows the shape of the distribution for kij with threevalues of ˛ revealing, in general, that on any given land parcel, mostother land parcels make only a small contribution to the populationgrowth of N. trichotoma. However, as ˛ increases it becomes morelikely that some values of kij will be large showing that it is possible,in some regions of parameter space, for other land parcels in theregion to have a significant effect on their neighbours. Fig. 5b showsthe distribution of kij with optimal parameter estimates for ˛ andq and this is discussed in Section 3.1.
Model input distributions and their parameter descriptions andestimates are given in Table 1.
3. Results
3.1. Model parameterisation
The model was parameterised for the Hurunui district of NorthCanterbury, New Zealand, by taking a sample set of values from thedistributions for carrying capacity, local per capita growth rate andgrubbing rate (Eq. (4), (6) and (7), respectively). This was repeated100,000 times (M = 100 land parcels and 1000 trials) and the result-ing model steady state distribution of plant densities (representedusing a relative frequency histogram with logarithmic binning andlog–log scale) was compared to the corresponding distribution of
the data (Fig. 6a) in the least squares sense for various combina-tions of q (year−1) and ˛ (km2). Each combination gives rise to onestochastic realisation of the steady state distribution. Repeat real-isations improved accuracy and the best fit values for the spreadrate distribution were estimated to be q = 6.7 × 10−4 year−1 and
A. James et al. / Ecological Modelling 222 (2011) 1030–1037 1035
1e−06 1e−04 0.01 11e−06
1e−04
0.01
1
100
1e+04
Spread parameter kij year−1
Rel
ativ
e fr
eque
ncy
aα=1
α=10
α= 0.1
1e−06 1e−05 1e−04 1e−031e−06
1e−04
0.01
1
100
1e+04
1e+06
Spread parameter kij year−1
Rel
ativ
e fr
eque
ncy
b
Fig. 5. The distribution of the spread parameter kij (year−1) for N. trichotoma asscmp
˛ms
tuotpTtidtgwcptstvnl
i
0.01 0.1 1 10 100 10001e−06
1e−05
1e−04
0.001
0.01
0.1
1
Density (plants ha−1)
Rel
ativ
e fr
eque
ncy
aDataBest ¯tr 0= .2
10 100 1e+03 1e+04 1e+051e−06
1e−04
0.01
Plants per property
Rel
ativ
e fr
eque
ncy
bData
Best fit
Fig. 6. Comparison of best fit heterogeneous model results with the Hurunui dataset for N. trichotoma for: (a) plant density (plants ha−1); (b) plants per land parcel.Logarithmic binning and a log–log scale are used and the leftmost bin includes the
pecified in Eq. (8). Logarithmic binning and a log–log scale have been used forlarity: (a) the effect of the shape parameter ˛ (km2) with q = 1.0 (year−1); (b) theodel enables the estimation of the spread distribution shown here (with optimal
= 0.36 km2 (Table 1). Note that standard derivative-based opti-isation methods do not work here due to the stochastic nature of
ampling.Fig. 6a shows a typical simulation steady state density distribu-
ion predicted by the best fit model. Although the fitting proceduresed only the density distribution data, the resulting distributionf plants per land parcel shown in Fig. 6b is also a good matcho the data within the data range. A small number of land parcelsredicted by the model have plant numbers near carrying capacity.his is not seen in the data and this discrepancy is addressed in Sec-ion 3.3.2. The best fit values predict a spread distribution as plottedn Fig. 5b. In this particular stochastic realisation the average of theistribution i.e. the expected value of kij is E(kij) = 1.6 × 10−5 year−1
hus the total annual expected number of new plants arising on aiven land parcel each year from all other land parcels (≈M × E(kij)here M = 100 land parcels in this example) is of the order 10−3. By
omparison the expected annual local per capita growth is r = 0.1lants per land parcel year−1, 4 orders of magnitude greater thanhe expected value of kij. Thus at steady state, on average, the mostignificant contribution to population growth comes from withinhe land parcel. Fig. 5a and Eq. (8) indicate that with the best fit
2
alue of ˛ = 0.36 km , there is potential for a small but significantumber of plants to originate from seeds dispersed from plants on
and parcels further away.The correlation of the model solution between the density of
ndividual land parcels and their individual parameter values was
zero counts. Also shown in (a) is the change in solution of the model resulting froman increase in the mean annual local per capita growth of N. trichotoma (from r =0.1 to 0.2 year−1). The parameter values are those in Table 1.
examined using a linear model and corresponding R2 value. Thisshowed that there is no connection between the density of plantson a land parcel and either the size of that land parcel or the totalspread rate from all other land parcels; R2 < 10−4 in both cases.However there is a small but significant correlation between theplant density on a land parcel and the local population growthrate (R2 ≈ 0.2) and between plant density and the grubbing rate(R2 ≈ 0.3).
If mean rates of the parameters were used in the homo-geneous model (r = 0.1 year−1, g = 0.43 year−1, C = 430 × 35,000plants, k = 1.6 × 10−5 year−1 and M = 100) the stable solution wouldbe the zero solution as the grubbing rate exceeds the critical rate i.e.g > r + (M − 1)k for these parameter values. By comparison, a typicalsimulation of the best fit heterogeneous model over the data rangegives a median per land parcel density of 5.9 plants ha−1, whichis very close to the measured value of 5.8 (Fig. 3). The predictedmedian number of plants per land parcel is approximately 1504whereas the data has a median of 375 plants.
3.2. Sensitivity analysis
The model was tested for robustness by making changes to thevalues and distributions of its inputs (Table 1) and by consideringthe effect that these changes have on the steady state distributionof plant density.
1036 A. James et al. / Ecological Modell
0.01 1 1001e−06
1e−04
0.01
1R
elat
ive
freq
uenc
yDataBest fit g~U(0,1) g~U(0.25,0.75)
Fs
ctoWio
vvrp
hdrwitHr
fNatfiHfi
3
th
3
≈W5gUeUtbdt
Density (plants ha−1)
ig. 7. The effects of different uniform grubbing distributions on the predictedteady state density distribution of the density of N. trichotoma (plants ha−1).
When changing the number of land parcels (M) in the model,are was taken to avoid changing the effect of other terms. Whenhe spread rate is scaled with M, i.e. q/M is kept constant the effectf different numbers of land parcels is minimal provided M > 50.hen the spread rate is not scaled in this way then the effect of
ncreasing the number of land parcels is commensurate with thatf increasing the spread rate.
Changing the carrying capacity or the mean land parcel size hasery little effect on the predicted distribution. The solution is alsoery robust to changes in the distribution of land parcel sizes. Thisobustness is a consequence of the fact that most land parcels havelant densities well below their carrying capacity.
By contrast, changing the mean annual local per capita growthas a more substantial effect. Fig. 6a shows the difference in pre-icted distribution for r = 0.1 year−1 (the original best fit value) and= 0.2 year−1. As would be expected, there are fewer land parcelsith low levels of N. trichotoma when the mean local growth rate
s increased. There is a strong, almost linear relationship betweenhe expected N. trichotoma density and the mean local growth rate.owever, changing the distribution of the local per capita growth
ate has only a very small effect.By further contrast, changing the distribution of the spread rate
rom the Gaussian (Eq. (8)) has a substantial effect on the predicted. trichotoma distribution. Other candidate distributions for thennual spread rate parameter, kij, were tested including exponen-ial, gamma and constant. In each case the solution distribution wastted using least squares and the best fit parameters were chosen.owever, the alternative distributions all gave substantially worsets to the data.
.3. Grubbing rate distributions
In the future, more realistic formulations of the model (see Sec-ion 4) could be applied using the methodology above to investigateow best to redistribute the grubbing effort across land parcels.
.3.1. Comparing grubbing strategiesCurrently the model assumes a mean grubbing rate of
43% year−1 via a best fit uniform distribution U(0, 0.86) (Fig. 4c).hat would be the consequences of increasing the mean to
0% year−1? In theory this could be done using a variety of strate-ies including: (1) increasing all rates to a uniform distribution(0, 1) where all grubbing rates between 0 and 100% year−1 arequally probable or (2) increasing rates to a uniform distribution
(0.25, 0.75) where no land parcels grub less than 25% or more
han 75% year−1 but all rates within these limits are equally proba-le. Fig. 7 shows the effect of these two strategies on the predictedensity distribution. Strategy 1 (U(0, 1)) results in little change tohe steady state distribution. By contrast, strategy 2 (U(0.25, 0.75)),
ing 222 (2011) 1030–1037
eliminates zero and low grubbing rates across all land parcels andresults in a substantial change to the steady state distribution witha substantial drop in the model median density from approximatebest fit values of 5.9 to 0.17 plants ha−1. There are fewer land parcelswith high N. trichotoma densities and more land parcels with verylow densities.
3.3.2. Density dependent grubbing ratesAlthough a good match between model plant numbers per land
parcel and the data is evident (Fig. 6b) there is a small but significantpercentage of model land parcels, mainly those with a grubbing rateclose to zero, with N. trichotoma plant numbers close to carryingcapacity. In reality any land parcel that was seen to be reaching suchhigh densities would be subject to external measures to counteractthis. This can be included in the model as a feedback mechanism inthe grubbing term which allows the rate to change depending onthe level of N. trichotoma reached.
4. Discussion and conclusion
In this paper a generic mathematical meta-population modelfor the growth, spread and control of an invasive plant has beendeveloped and applied to N. trichotoma using parameters estimatedfrom experimental data from the Hurunui district, North Canter-bury, New Zealand. A number of key results have arisen from thisinvestigation.
Firstly there is a marked difference between the heterogeneousmodel, where parameter values are sampled from distributions,and the corresponding homogeneous model where only meanparameter values are used. The difference between model out-comes of stochastically sampled parameters and their mean valuesis well documented and known as Jensen’s inequality (see forexample Ruel and Ayres, 1999). This highlights the importance ofincluding heterogeneity in meta-population models. When con-sidering a complex system where there are many contributingindividuals it is imperative to have as much information aboutthese as possible. The homogeneous model, at the mean parametervalues, predicts a zero steady state solution since the mean grub-bing rate is above that needed to eradicate N. trichotoma, while theheterogeneous model, using the best fit parameters from Table 1,predicts a non-zero equilibrium implying species persistence underthe current grubbing strategy.
Secondly the model gives the first known estimate of the cur-rent distribution of the rates of spread of N. trichotoma between landparcels (Fig. 5b). It predicts that the expected annual local per capitagrowth is large compared to the expected annual rate of spreadfrom another land parcel. At the current low densities of the weedthis implies that the total expected number of plants received fromall other land parcels is very small compared to within land parcelpopulation growth. In the Hurunui district there is much discussionbetween land owners on the relative efficacy of their neighbours’(both near and far) grubbing technique. This study suggests that,while spread between land parcels is important the majority of aland parcel’s N. trichotoma plants originate from itself. This is shownby the small expected value of the total spread rate onto a landparcel in relation to the expected local growth rate. It is also appar-ent in the lack of correlation in the solution between density andtotal spread rate. We note that although this spread between landparcels appears highly insignificant by these measures, the modelbecomes almost nonsensical without it; with no coupling between
land parcels, each land parcel will either reach carrying capacityor have zero density depending on its local per capita growth andgrubbing rates.
Thirdly, the robustness of the model was tested by consideringthe effects of varying the inputs on the steady state density distribu-
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ion. The model’s steady state distribution was sensitive to changesn the local mean growth rate and the spread rate distribution. Theatter distribution is currently unknown. To redress this, a seriesf experiments is planned in which the distances that the seed-earing panicles of N. trichotoma disperse from the parent plant, inelation to wind speed and direction, will be estimated.
Interestingly the steady state density distribution was not sensi-ive to changes in the local per capita growth rate distribution or theand parcel size distribution and the mean value of each could beubstituted with no obvious impact. We note that in reality chang-ng the size distribution would affect the spread distribution. Thiseads to a natural extension of the model by incorporating farmoundaries as model inputs and hence carrying capacity and dis-ersal are chosen according to the geographical location of each
and parcel.Finally the model enables the comparison of the effects of alter-
ative grubbing strategies and it is envisaged that it will form theasis of a management decision-making tool for N. trichotoma inhe future. The model presented here is a stepping stone in thatirection. Four key areas have been identified for enhancement ofhe model:
Dispersal of panicles. Here, in lieu of data, we have used the modelto estimate the spread parameter distribution (Eq. (8)). This esti-mate requires experimental validation.Spatial settings. The current model selects a value for each param-eter for every land parcel at random from a specified distribution.Utilising actual land parcel boundaries would enable exact areas,distances between land parcels and carrying capacities for eachland parcel to be considered.Real landscape information. Currently the locations of N. tri-chotoma infestations are recorded electronically in a land parcelbased database with the entire land parcel being marked‘infested’ if N. trichotoma is present anywhere within its bound-ary. However, N. trichotoma plants are found in higher densitieson northerly-facing slopes and are relatively rare on south-facingslope and flat land (Bourdôt and Saville, 2005). With advancesin geographic information systems it may be possible to incor-porate these real landscape features and have the distributionsinfestation-based rather than land parcel based.Plant size dependent grubbing rates. In the current model all plantsin a population are considered to be equally likely to be grubbedregardless of their size. This is known not to be true; lowergrubbing rates occur among very small (easily missed) and verylarge (mistaken for native tussock) plants (Verkaaik et al., 2006).A plant-size-structured local population model, currently beingdeveloped, will be incorporated into the spatial model to enablemore realistic simulation of grubbing. In addition, this mod-ification will allow an individual plant’s contribution to localpopulation growth to depend upon its size, as occurs in reality.
There are currently numerous modelling techniques that can bepplied to invasive species (for example see the reviews of Hastingst al., 2005; Katul et al., 2005 and references therein). We have cho-en a meta-population model because it fits naturally to the spread
f an invasive plant between land parcels. The model presentedn this paper has been applied to N. trichotoma populations in theurunui district of New Zealand. N. trichotoma is also a seriousasture weed in Australia and South Africa and the model coulde applied equally well in these countries. Since it is generic, it
ng 222 (2011) 1030–1037 1037
is applicable to any other invasive plant species exhibiting meta-population dynamics involving local per capita growth, spread andlocal removal of plants.
Acknowledgements
We thank Laurence Smith, Environment Canterbury, Amberleyand Ryan Elley, Environment Canterbury, Christchurch for theirvaluable contributions towards the manuscript. We also thank theNew Zealand Foundation for Research, Science and Technology forfunding under the Undermining Weeds programme (C10X0811),the New Zealand Institute of Mathematics and its Applications forawarding a post doctoral fellowship to R.B., and the University ofCanterbury for awarding a post doctoral fellowship to B.B.
Under Environment Canterbury’s Regional Pest Management Strategy – Year 10(2006–2007).
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