Comparison of Two Types of Metapopulation Models in … papers/Lizard... · 1102 Conservation Biology, Pages 1102–1113 Volume 15, No. 4, August 2001 Comparison of Two Types of Metapopulation

1102

Conservation Biology, Pages 1102–1113Volume 15, No. 4, August 2001

Comparison of Two Types of Metapopulation Models in Real and Artificial Landscapes

D. GRANT HOKIT,* BRADLEY M. STITH, and LYN C. BRANCH

Department of Wildlife Ecology and Conservation, University of Florida, Gainesville, FL 32611, U.S.A.

Abstract:

Application of metapopulation models is becoming increasingly widespread in the conservation ofspecies in fragmented landscapes. We provide one of the first detailed comparisons of two of the most com-mon modeling techniques, incidence function models and stage-based matrix models, and test their accuracyin predicting patch occupancy for a real metapopulation. We measured patch occupancies and demographicrates for regional populations of the Florida scrub lizard (

Sceloporus woodi

) and compared the observed oc-cupancies with those predicted by each model. Both modeling strategies predicted patch occupancies withgood accuracy (77–80%) and gave similar results when we compared hypothetical management scenariosinvolving removal of key habitat patches and degradation of habitat quality. To compare the two modelingapproaches over a broader set of conditions, we simulated metapopulation dynamics for 150 artificial land-scapes composed of equal-sized patches (2–1024 ha) spaced at equal distances (50–750 m). Differences inpredicted patch occupancy were small to moderate (

�

20%) for about 74% of all simulations, but 22% of thelandscapes had differences of

�

50%. Incidence function models and stage-based matrix models differ in theirapproaches, assumptions, and requirements for empirical data, and our findings provide evidence that thetwo models can produce different results. We encourage researchers to use both techniques and further exam-ine potential differences in model output. The feasibility of obtaining data for population modeling varieswidely among species and limits the modeling approaches appropriate for each species. Understanding differ-ent modeling approaches will become increasingly important as conservation programs undertake the chal-lenge of managing for multiple species in a landscape context.

Comparación de Dos Tipos de Modelos de Metapoblaciones en Paisajes Reales y Artificiales

Resumen:

Cada vez es mayor la aplicación de modelos de metapoblaciones para la conservación de especiesen paisajes fragmentados. Proporcionamos una de las primeras comparaciones detalladas de dos de las técni-cas más comunes para modelar, los modelos de función de incidencia y los modelos demográficos basados enetapas, y probar su precisión en una metapoblación real. Medimos la tasa de ocupación de fragmentos y lastasas demográficas de poblaciones regionales de la lagartija

Sceloporus woodi

y comparamos la ocupaciónobservada con la predicha por cada modelo. Ambas estrategias de modelaje predijeron la ocupación de frag-mentos con buena precisión ( 77–80%) y dieron resultados similares cuando comparamos escenarios demanejo hipotéticos que involucraban la remoción de fragmentos de hábitat claves y la degradación de la cal-idad del hábitat. Para comparar las dos aproximaciones al modelaje en un conjunto de condiciones más am-plio, simulamos la dinámica metapoblacional en 150 paisajes artificiales compuestos de fragmentos delmismo tamaño (2–1024 ha) separadas por la misma distancia (50–750 m). Las diferencias en la ocupaciónde fragmentos predicha fueron pequeñas a moderadas (menos de 20%) en aproximadamente 74% de todaslas simulaciones, pero 22% de los paisajes tenían diferencias que excedían el 50%. Modelos de función de in-cidencice y modelos de matriz debido a la naturaleza artificial de estos paisajes y la ausencia de datos decampo, consideramos que estos resultados proporcionan evidencia preventiva de que los dos modelos puedenproducir resultados distintos. Discutimos como los modelos de función de incidencia y los modelos demográ-ficos basados en etapas son distintos en sus aproximaciones, supuestos y requerimientos para datos empíri-cos, y sugerimos que los investigadores utilicen ambas técnicas y comparen resultados cuando sea posible. La

*

Current address: Department of Biology, Carroll College, Helena, MT 59625, U.S.A., email [email protected] submitted April 10, 2000; revised manuscript accepted October 18, 2000.

Conservation BiologyVolume 15, No. 4, August 2001

Hokit et al. Comparison of Two Metapopulation Models

1103

Introduction

How a species responds to the heterogeneous distribu-tion of habitat and resources may strongly influence itslong-term persistence ( MacArthur & Wilson 1967; Levins1969; Simberloff & Abele 1976; Soulé 1987; Fahrig & Pal-oheimo 1988; Gilpin & Hanski 1991). Whether a particu-lar species is likely to persist in a given landscape is animportant question from the standpoints of both theoryand practice. Conservation biologists are increasingly re-lying on metapopulation models to address such ques-tions (e.g., Hanski & Thomas 1994; LaHaye et al. 1994;Turner et al. 1995; Hanski et al. 1996; Wiens 1996; Han-ski & Simberloff 1997 ).

A variety of metapopulation models are now availablethat differ greatly in data requirements and model as-sumptions (Hanski 1999). Two popular but different typesof metapopulation models are the incidence functionmodel (Hanski 1994) and stage-based matrix models suchas Vortex, ALEX, and RAMAS (reviewed by Lindenmayer etal. 1995). Stage-based matrix models have been appliedto a broad range of species; examples include the SnailKite (

Rostrhamus sociabilis

; Beissinger 1995), FloridaScrub-Jay (

Aphelocoma coelurescens

; Root 1998; Brein-inger et al. 1999), Florida manatee (

Trichechus manatus

;Marmontel et al. 1997 ), and lower keys marsh rabbit(

Sylvilagus palustris hefneri

; Forys & Humphrey 1999).The weaknesses of stage-based matrix models have beenreviewed by Beissinger and Westphal (1998) and Groomand Pascual (1998). Incidence function models have beendeveloped for fewer species; examples include Floridascrub plants (Quintana-Ascencio & Menges 1996) and sev-eral species of butterflies (Hanski 1999). Because of theincreasing popularity of the incidence function andstage-based matrix models, it is important for biologiststo understand the substantial differences between thesetwo approaches.

We provide one of the first detailed comparisons of in-cidence function and matrix approaches by modelingthe metapopulation dynamics of the Florida scrub lizard(

Sceloporus woodi

) within a large study area with natu-rally occurring habitat patches. We examined the rela-tive merits and limitations of the incidence functionmodel and a stage-based matrix model by comparingtheir data requirements and assumptions and their pre-dictive accuracy based on patch-occupancy data. Wealso compared predictions of the two models for differ-ent management scenarios within our study area by sim-

ulating degradation of habitat quality and removal of keyhabitat patches. To extend our comparison to otherlandscape configurations, we simulated a series of 150artificial landscapes varying in patch size and isolationand evaluated the congruence of predicted patch occu-pancy for the two models.

Methods

Study Site, Species Biology, and Patch Data

We conducted our study at Avon Park Air Force Range(15 km east of Avon Park, Florida), which contains oneof the largest remaining areas of scrub (2300 ha) in cen-tral Florida. Florida scrub is found on relic sand dunes ofthe central ridge of the Florida peninsula and in strandsalong more recent coastal sand dunes (Christman 1988;Fernald 1989; Meyers 1990). This unique system is natu-rally fragmented into an archipelago of habitat islands sur-rounded by more mesic and hydric habitats unsuitablefor obligate scrub organisms. The scrub lizard is en-demic to scrub habitat and some open sandhill habitatsof south and central Florida ( Jackson 1973

a

, 1973

b

; De-Marco 1992).

We visually delineated all scrub patches on Avon ParkAir Force Range through soil, topographic, and vegetationmaps and infrared aerial photographs (

n

�

95 patches).Boundaries of patches were drawn on aerial photos anddigitized on scanned, georeferenced images of these pho-tographs. To analyze the landscape at a spatial scale rele-vant to scrub lizards, we delineated regional clusters ofpatches likely to be isolated demographically from oneanother. Using a geographic information system (GIS),we generated dispersal buffers around each scrub patchthat represented the estimated maximum dispersal dis-tance of scrub lizards (750 m, Hokit et al. 1999) andgrouped patches with overlapping buffers into clusters( Fig. 1). For each scrub patch in each cluster, we mea-sured total area of the patch, percent cover of bare sandyhabitat (a measure of habitat quality for scrub lizards;Hokit et al. 1999), and edge-to-edge distance to all otherpatches within the same cluster (for details of GIS proce-dures, see Branch et al. 1999; Hokit et al. 1999). To de-termine the presence or absence of scrub lizards, weconducted visual searches annually (1994–1998) in allscrub patches, and more often when no lizards werefound (two to five times each year). Only two of the six

factibilidad de obtener datos para modelar poblaciones varía ampliamente entre especies, y limita las aprox-imaciones de modelaje apropiadas para cada especie. El entendimiento de las distintas aproximaciones almodelaje será cada vez más importante a medida que los programas de conservación acepten los retos de

manejar múltiples especies en el contexto del paisaje.

1104

Comparison of Two Metapopulation Models Hokit et al.


clusters (referred to as north and south clusters) con-tained scrub lizard populations, although habitat was ad-equate in other clusters ( Hokit et al. 1999). In 1994 wefound scrub lizards in 17 of the 33 patches in the southcluster and in 9 of 29 patches in the north cluster ( Fig.1). Between 1994 and 1998, we recorded two coloniza-tion events and six extinction events.

Incidence Model

MODEL

STRUCTURE

The incidence function model, described in detail byHanski (1994, 1999), relies on a “snapshot” of the pres-ence or absence of a species in a set of habitat patches.The modeling objective is to estimate patch-specific col-onization and extinction probabilities for a particularnetwork (cluster) of patches and then use these esti-mates to make predictions about patch occupancy, tran-sient dynamics, and regional population persistence.The primary assumptions of the Hanski model are that(1) the observed cluster of patches is in a dynamic equi-librium of colonization and extinction; (2) local (within-patch) demographic processes occur quickly relative toregional dynamics and may be ignored; (3) patch occu-pancy can be determined for discrete intervals (e.g.,patches are either occupied or unoccupied for the unit

time interval); (4) extinction (

E

i

) is a function of patchsize; (5) colonization (

C

i

) is a function of the number ofmigrants and the propagule size required to establish apopulation; (6) within discrete time intervals, extinctionoccurs before colonization; ( 7 ) if

E

i

and

C

i

are relativelyconstant, then the probability of patch occupancy (

J

i

)equals

C

i

/

C

i

�

E

i

.We modeled the patch-specific extinction probability

(

E

i

) for scrub lizards as follows (Hanski 1994; Hanski et al.1996):

(1)

where

A

i

is the total area ( ha) of patch

i

,

�

is the proba-bility that a population in a patch of unit size (e.g., 1 ha)will go extinct, and

x

is a coefficient describing the rela-tionship between patch size and extinction probability.Because scrub lizards prefer open, sandy habitat, we re-placed

A

i

in equation 1 with the “effective area” (

A

i

�

)given by

Q

i

A

i

/

Q

*, where

Q

i

is the percentage of open,sandy habitat estimated for patch

i

and

Q

* is the percent-age of open, sandy habitat for the patch with the highestproportion of bare, sandy habitat (Hanski 1994).

We assumed that patch-specific colonization probabil-ity (

C

i

) depends on the distance to other patches withinthe same cluster occupied by scrub lizards and on the re-spective areas of occupied patches (which are associ-ated with population size and thus, presumably, withthe number of potential migrants). Thus,

C

i

is defined as

(2)

where

y

�

is a parameter that describes the colonizationability of the species and

S

i

is a patch-specific isolationparameter defined as

(3)

where

p

j

is 0 for unoccupied patches and 1 for occupiedpatches,

�

is a coefficient describing the relationship be-tween migrant survival and migration distance,

d

ij

is thedistance (m) from patch

i

to patch

j

, and

A

j

�

is the effec-tive area of patch

j.

To estimate coefficient

�

, we usednonlinear regression to fit the equation

P

�

exp(

�

d

)to the observed proportion of occupied patches (

P

) as afunction of distance (

d

) to the nearest occupied patch(Hokit et al. 1999). We then estimated the

S

i

values us-ing an analog program written in BASIC.

Following Hanski (1994), and provided that a particu-lar regional population complies with the given assump-tions, the patch-specific probability of occupancy (

J

i

)may be defined by the model

(4)

Alternatively, the patch-specific probability may be de-fined by

(5)

Ei min µ Aix, 1⁄[ ],=

Ci 1 1 y ′ Si⁄( )2+( ),⁄=

Si Σpjexp αdij–( )A′j ,=

Ji 1 1 y′ Si⁄[ ]2+( )µ Aix⁄( ).⁄=

Ji 1 1 µ′ SizAi

x⁄( )+( ),⁄=

Figure 1. Map of scrub patches occupied or unoccu-pied by Florida scrub lizards on Avon Park Air Force Range in 1997. Dispersal buffers of 750 m (estimated maximum dispersal distance of scrub lizards) are drawn around each scrub patch to form clusters of patches that may be linked demographically. North (N) and south (S) clusters of patches were modeled as separate populations. Three patches were removed ( R) for a subset of simulated management scenarios.


Hokit et al. Comparison of Two Metapopulation Models 1105

where z is a constant that determines the shape of thefunction relating the number of migrants to the proba-bility of successful colonization and �� is a compositevariable described by �� y�2 (Hanski 1994). Equation5 incorporates “rescue effects” in which local popula-tions may go extinct but then be recolonized within anydiscrete time interval (Hanski 1994).

PARAMETER ESTIMATION

To estimate patch-specific colonization and extinctionprobabilities, the unknown parameters in equation 1 (�and x) and equation 2 ( y�) must be estimated by meansof either equation 4 or 5. We fit both equations to theobserved patch-occupancy data for scrub lizards. We re-placed the unknown Ji’s by the observed occupancies( pis) and, using maximum likelihood regression (Neteret al. 1989), we fit both equations to pooled data fromthe north and south clusters. To assess which equation(4 or 5) best fit the data, we compared the alternativesto a logistic regression model that included only Ai� data.Patch area was positively associated with scrub lizardpatch occupancy ( p � 0.001) in the logistic model.Lacking spatial information, however, the logistic regres-sion model did not include effects due to patch locationand provided a null model to compare with the alterna-tive incidence function models (Hanski et al. 1996). Wecompared the models using likelihood ratio tests and es-timates of the minimum negative log-likelihood function(Neter et al. 1989).

Both alternatives fit the data better than the logisticmodel ( Table 1). Equations 4 and 5 could not be com-pared with likelihood ratio tests because of the lack ofdegrees of freedom. Equation 5 had a lower log-likeli-hood function, however, suggesting a better fit. There-fore, we used the parameter estimates from equation 5in all subsequent analysis.

To estimate the extinction parameter �, we first esti-mated y� using data on patch-occupancy turnover (colo-nization and extinction) from 5 years of observationsand the equation described by Hanski (1994):

(6)T Σ1 Si2

y′2 Si2

1 pi–( ) µ′pi Aix⁄+[ ],+⁄=

where T is the total number of turnover events. Usingequation 6, the total number of observed turnover events(8), patch data, and the estimates of �� (1.258) and x(0.974), we numerically determined the value of y� (4.04)with a program written in BASIC. We then estimated thevalue of � (0.076) from � � ��y�2 ( Hanski 1994).

Stage-Based Matrix Model

Our matrix model, developed with RAMAS/metapop (Ak-çakaya 1994), incorporated a stage-matrix approach formodeling local (within-patch) demography and a dis-tance function for modeling migration rates betweenpatches. Demographic and dispersal parameters werebased on data from an extensive field study of scrub liz-ards that included mark-recapture on eight trappinggrids (each 1 ha) sampled at 1-month intervals for 2 years(Branch et al. 1996). Our model assumptions were as fol-lows: (1) a single sex (females) was modeled; (2) dis-persal between patches was simulated with a simple dis-tance-weighted function; (3) density dependence wasmodeled with a population ceiling; (4) demographic sto-chasticity of survival was modeled with a binomial func-tion, stochasticity of fecundity with a poisson function,and environmental stochasticity with annual standarddeviates (Akçakaya 1994); (5) potentially important fac-tors for which no data exist were ignored (e.g., catastro-phes, correlation of environmental stochasticity).

Because scrub lizards rarely survive more than 2 years(Hartmann 1995; Branch et al. 1996), we described localpopulation demography using a three-stage matrix ( Ta-ble 2). We defined stage 1 as occurring from the time ofhatching ( June–November) to the following May, the be-ginning of the lizards’ first reproductive season. Stages 2and 3 occurred during the lizards’ first and second activereproductive seasons, respectively. Because we couldnot accurately age lizards after they reached 45 mm insnout-vent length ( Jackson & Telford 1974; DeMarco1992), we assumed that fecundity and survival rateswere equivalent for these stages ( Table 2). In other ar-eas, females oviposite an average of four eggs per clutchand produce an average of three clutches per year( Jackson & Telford 1974). Egg mortality has not beenexamined for scrub lizards but has been estimated as5–10% for other Sceloporus (e.g., Overall 1994). Based onthese studies, we estimated that females in large patchessuccessfully produce 11 hatchlings per reproductive sea-son. Because recruitment in small patches (�7 ha) is44% lower than recruitment in large patches (Branch etal. 1996), we estimated that females in small patchesproduce 6.16 hatchlings per reproductive season.

To estimate stage 1 survivorship, we calculated survi-vorship to the following May for each monthly cohortfrom June through November in small and large patches.We then averaged survivorship across the cohorts forboth small and large patches (Table 2; Branch et al. 1999).

Table 1. Results of comparisons between alternative versions of the incidence function model and the simple logistic model for patch area.*

Model np LFModels

compared LRT 2 df p

(1) Equation 4 3 41.02 1 vs. 3 27.00 1 �0.001(2) Equation 5 3 34.14 2 vs. 3 40.76 1 �0.001(3) Logistic model 2 54.52

*Abbreviations: np, number of estimated parameters; LF, minimumvalue of the negative log-likelihood function; LRT 2, likelihood ratiochi-square statistic; df, degrees of freedom for LRT 2.

1106 Comparison of Two Metapopulation Models Hokit et al.


We estimated stage 2 and 3 survivorship rates by multi-plying the monthly survivorship estimates ( large patches,0.83%; small patches, 0.79%; Branch et al. 1996) over a12-month annual cycle ( Table 2).

Because we lacked long-term data, we did not includethe effects of catastrophes or correlations in environ-mental stochasticity among patches. We assumed thatpatches were at carrying capacity and lizards had a sta-ble age distribution at the start of each simulation. Weestimated the carrying capacity of each patch by multi-plying the effective area by the average density estimate(54 lizards per ha for patches of �7 ha; 82 lizards per hafor patches of �7 ha) determined from mark-recapturestudies ( Branch et al. 1996). The mean individual growthrate of females is influenced by density ( Branch et al.1996). Therefore, we assumed that ceiling-type densitydependence operates in all patches (Akçakaya 1994).

Our estimates for scrub lizard dispersal characteristicswere speculative because there are no data on the dis-persal rates of scrub lizards or of most other small-bod-ied lizards. Clobert et al. (1994) found that more than50% of juvenile common lizards (Lacerta vivipara) dis-persed farther than the size of one adult home rangewithin a habitat patch, but they failed to document dis-persal between habitat patches. Similarly, of more than1300 scrub lizards captured on 1-ha sampling plots, werecaptured 10% of the lizards between 50 and 100 mfrom the edge of the sampling plots but did not docu-ment interpatch dispersal ( Branch et al. 1996). For mod-eling purposes, we assumed that 10% of stage 1 lizardsdisperse from each occupied patch. Because adults haverelatively stable home ranges ( Branch et al. 1996), weassumed that only stage 1 individuals migrate betweenscrub patches and that migrants disperse among allpatches within 750 m of the source patch. To providedistance-weighted estimates for the pairwise migrationmatrix in RAMAS/metapop, we used the function

(7)

where mij is the annual proportion of migrants frompatch i to patch j, � and dij are as in the incidence func-tion models above, ni is the total number of migrantsfrom patch i (estimated as 10% of the juvenile popula-tion in patch i at carrying capacity), t is the number of

mij exp αdij–( ) ni t⁄( ) Ni,⁄=

patches within 750 m of patch i, and Ni is the abun-dance of patch i at carrying capacity.

Assessment of the Accuracy of Models in PredictingPatch Occupancy

We simulated the transient dynamics of patch occu-pancy by scrub lizards in the north and south clusters ofscrub patches (Fig. 1) with both models. We iterated theRAMAS model for 100 years, starting with all patches oc-cupied, and for 100 replications. For the incidence func-tion approach, we used an iterative program to generatestochastic extinction and colonization events by com-paring randomly generated numbers with estimates (us-ing equations 1 and 2) of the patch-specific extinctionand colonization probabilities per unit time interval (1year). We iterated the model for 1500 years, startingwith all patches occupied and replicated this procedurefor 100 simulations. The different time scales for thestage-based matrix model and incidence function modelreflect differing rates of the two modeling techniques inreaching equilibrium. To assess the performance of themodels, we (1) observed whether the models stabilizedat an equilibrium value for the proportion of occupiedpatches, (2) compared model predictions for the pro-portion of occupied patches with the observed propor-tion of occupied patches in the field, and (3) comparedthe observed versus the predicted patch occupancy forspecific patches using Fisher exact tests.

Model Sensitivity

We performed sensitivity analyses to assess how sensi-tive the models were to parameter estimates. In the inci-dence function model, the estimated parameters used inequations 1 and 2 were increased individually by 25%. Inthe stage-based matrix model, fecundity, survivorship,and migration parameters were increased individually by25%. For each model, 100 replicate simulations of tran-sient dynamics were performed for each parameterchange to estimate the proportion of occupied patches.To obtain sensitivity values, we divided the percent changein the model output by the percent change in the inputparameter.

Comparison of Simulated Management Scenarios

An important application of models in conservation is tounderstand the potential effects of environmental change.Using both models, we generated 50-year projections ofthe proportion of occupied patches for the north andsouth clusters of scrub patches under three scenarios.For scenario 1, we assumed that habitat quality wouldremain constant for the next 50 years. For scenario 2,we included a 6% decrease in habitat quality annually

Table 2. Stage matrices for large and small (�7 ha) patches used to parameterize the matrix model.

Large patch Small patch

stage stage

1 2 3 1 2 3

Fecundity 0 11 11 0 6.16 6.16Survivorship 0.18 0 0 0.12 0 0Survivorship 0 0.11 0.11 0 0.06 0.06



over the 50-year interval to simulate the effects of firesuppression. Our estimate of change in habitat qualitywas based on estimates of the decrease in sandy habitatin scrub patches in the absence of burning from a se-quence of aerial photographs of Avon Park Air ForceRange over six decades ( Branch et al. 1996). The de-crease in habitat quality was modeled as a decrease in ef-fective patch area in the incidence function model andas a linear decrease in carrying capacity in the stage-based matrix model. For scenario 3, we included an an-nual decrease in habitat quality and removed from thesimulation two large patches from the south and onelarge patch from the north as if the patches had beeneliminated from the landscape ( Fig. 1). Because we as-sumed that initial abundance was at carrying capacityfor all occupied patches, dynamics were highly variablefor the first 40 years of simulations in the stage-basedmatrix model. We allowed this model to run for 90 yearsand then eliminated the first 40 years. Because of the ar-bitrary nature of this cut-off point, we could not directlycompare the trajectories of the stage-based matrix modeland the incidence function model. Our goal for this setof simulations was a qualitative comparison of the twomodels.

Comparison within Artificial Landscapes

To compare the two modeling approaches across a rangeof landscape configurations, we applied the models toartificial landscapes composed of 20 patches. All patcheswere of equal size and distance from one another, andwe systematically varied patch size from 2 to 1024 ha atgeometric increments (2, 4, 8, . . .) and isolation from 50to 750 m at 50-m increments. To simplify model setup,and following the assumption of Levins’ (1969) originalmetapopulation model, we assumed that all patcheswere the same distance from one another. For each sim-ulation, we used the same parameters as in our naturallandscape model and started with all patches occupied.We then modeled the transient dynamics of patch occu-pancy as described in the section on model accuracy.We repeated this simulation procedure for various area-by-isolation combinations and compared the proportionof patches predicted to be occupied by the incidencefunction model and the matrix model.

To assess how model parameters may contribute todifferences between the two models, we conducted ad-ditional simulations for these two area-by-isolation com-binations. We systematically manipulated each of thefive parameters for the stage-based matrix model and re-peated simulations in an attempt to achieve congruencewith the incidence function model. We repeated thisprocedure for the four parameters of the incidence func-tion model in an attempt to achieve congruence withthe matrix model. Our purpose was not to complete acomprehensive analysis of all combinations of all param-

eters but rather to quickly assess whether congruencewas possible and which parameter manipulations wouldpotentially produce congruence. Inability to achieve con-gruence might indicate that differences were due to modelstructure rather than parameter values. If congruency couldbe achieved, we noted whether the altered parameter set-tings were plausible biologically. Differences between thetwo models might then be due to poor parameter estimatesfor one or both models.

Results

Comparison of Model Accuracy and Sensitivity in aReal Landscape

Both the incidence function and stage-based matrix mod-els approached a steady state for the predicted propor-tion of occupied patches for both the south and northclusters of patches (Fig. 2). The incidence function model

Figure 2. Transient dynamics of estimated patch occu-pancy (�SD) by scrub lizards for the south cluster us-ing (a) the incidence function model and ( b) the stage-based demographic model. Transient dynamics of the north cluster (not shown) exhibited similar properties. Different time scales reflect different rates required to reach equilibrium for the two modeling techniques.



predicted 17 occupied patches for the south cluster and11 occupied patches for the north cluster. The matrixmodel predicted 15 occupied patches for the south clus-ter and 11 occupied patches for the north cluster. Bothmodels compared well with the observed patch-occu-pancy data. In most years, 17 patches were occupied inthe south cluster and 9 patches were occupied in thenorth cluster.

Both models also did well in predicting patch occu-pancy for specific patches in the south and north clus-ters (all Fisher’s exact tests, p � 0.001). The incidencefunction model correctly predicted patch occupancy for73% of the patches in the south cluster, 83% of thepatches in the north cluster, and 77% overall. The stage-based matrix model correctly predicted patch occu-pancy for 84% of the patches in the south cluster, 76% ofthe patches in the north cluster, and 80% overall. Thetwo models agreed on 74% of the overall patch occupan-cies. Agreement was weakest for patches of 2–12 ha( Fig. 3). The six largest patches had areas of �12 ha andwere always predicted to be occupied by both models.

The 23 smallest patches were �2 ha in area and were al-ways predicted to be unoccupied except for 3 that werewithin 100 m of a large patch.

Both models are insensitive to changes of 25% for mostparameter values ( Table 3); the incidence function modelwas somewhat less sensitive than the stage-based matrixmodel. None of the sensitivity values for the incidencefunction model were above 1, indicating that the changein the model output was less than the change in the pa-rameter input. The largest sensitivity value for the ma-trix model was 1.40 for stage 1 survivorship.

Comparison of Management Scenarios

Starting with patch occupancy data for 1995, both mod-els projected a relatively constant steady state for bothclusters for the next 50 years (Fig. 4). But simulationsthat incorporated the effects of fire suppression by annu-ally decreasing habitat area (incidence function model) orcarrying capacity (stage-based matrix model) resulted ina decline in patch occupancy (Fig. 4). Simulation ofpatch removal accelerated this decline. For each of thethree management scenarios, the two models producedsimilar population trajectories.

Comparison of Artificial Landscapes

Our comparison of the two modeling approaches for150 artificial landscapes showed relatively small dis-agreement (�20% difference in patch occupancy) for amajority (72%) of the configurations ( Fig. 5), but strongmodel disagreement (�50% difference in patch occu-pancy) occurred in two regions accounting for 22% ofthe configurations. Maximum model disagreement forthese two regions was centered at 2-ha patches sepa-rated by 50 m (small-near patches) and 32-ha patchesseparated by 750 m ( large-isolated patches; Fig. 5). Inthe small-near region, the incidence function model pre-dicted much higher patch occupancies than the stage-based matrix model. In the large-far region, the inci-

Figure 3. Predicted and observed patch occupancies for three patch-size classes on Avon Park Air Force Range.

Table 3. Analysis of the sensitivity of the incidence function and stage-based matrix models to changes in parameter estimates.

Model Parametera Sensitivity valueb

Incidence function model� � 25% 0.67x � 25% 0.37y� � 25% 0.74� � 25% 0.52

Stage-based matrix modelp 0.27� � 25% 0.20fecundity � 25% 1.04stage 1 survivorship � 25% 1.40stage 2–3 survivorship � 25% 0.32

aThe parameter p is the percentage of stage 1 migrants leaving a patch. All other parameters are as defined in the text.bChange in model output divided by the change in model input (25%).



Figure 4. Fifty-year projections of the estimated pro-portion of occupied patches (�SD) by scrub lizards for the south cluster (1) when habitat quality remains constant (circles), (2) when there is a constant annual decrease in habitat quality (triangles), and (3) when there is a constant annual decrease in habitat quality and elimination of two patches (squares). Projections were produced for both (a) the incidence function model and ( b) the stage-based matrix model and were similar for north cluster patches (not shown).

dence function model predicted much lower patchoccupancies than the matrix model.

Congruency Analysis of Artificial Landscapes

The congruency analysis for the small-near patch config-uration indicated that no single parameter of the stage-based matrix model could be realistically modified toproduce the high level of occupancy predicted by theincidence function model ( Table 4a). Fecundity andstage 1 survivorship could be manipulated individuallyto produce congruence but only at twice the fecundityrate (22 hatchlings per female) or 1.2 times the survivor-ship rate (0.22) observed for scrub lizards in large patches( Branch et al. 1996). A combination of 9.9 for fecundityand 0.16 for stage 1 survivorship resulted in congruencewith the incidence function model, however, and bothof these vital rates are realistically between field esti-mates for small and large patches. Congruence could notbe achieved by altering migration with realistic parame-

ter estimates. In our matrix model, even when morethan 50% of stage 1 lizards were allowed to migrate toneighboring patches or migrants were allowed to sur-vive for infinite distances (i.e., � � 0), migration did notrescue small neighboring patches from extinction. Incontrast, the rescue effect was evident for the incidencefunction model. Even when extinction was maximal(� � 10 or x � 4), patches were rescued by recoloni-zation. To obtain congruency between the two modelsby reducing dispersal in the incidence function model(i.e., to obtain 0% occupancy of small patches), survivor-ship of lizards dispersing 50 m or more could not ex-ceed 4% and turnover events per year could not exceed0.04 (i.e., � � 0.06, y� � 43). Field observations of patchturnover and scrub lizard movements suggest muchhigher turnover rates and more successful dispersal (Ho-kit et al. 1999).

Analysis of the landscape of large, isolated patches (32ha, 750 m apart) indicated that no single parameter ofthe incidence function model could be modified to pro-duce the high level of occupancy predicted by the ma-trix model ( Table 4b). But simultaneously decreasing �and increasing x, two parameters relating to the extinc-tion probability of a patch, resulted in output similar tothat of the stage-based matrix model ( Table 4b); this oc-curred with biologically feasible values of x (2.0 is justover 3 SE from our estimated value) and � (0.02, whichdecreases the probability of a 32-ha patch going extinctfrom 0.003 to 0.001). In contrast, the fecundity and sur-vivorship parameters in the matrix model had to be re-duced to values inconsistent with observations for scrublizards to achieve congruence with the incidence func-tion model ( Table 4b; Branch et al. 1996). A fecundity ofsix hatchlings per female would be almost half thatobserved for large patches, and stage 1 survivorship of0.11 would be less than the survivorship rate in smallpatches.

Discussion

Model Comparison in a Real Landscape

The two approaches to modeling metapopulation dy-namics differ substantially in their assumptions and datarequirements. Incidence function models use simple pres-ence-absence data, completely ignore demographic fac-tors such as mortality and fecundity, and do not track lo-cal population size; the primary measure is whether apatch is occupied or not. In contrast, stage-based matrixmodels track the number of individuals within eachpatch. Local patch dynamics are modeled by means ofdemographic factors for each life stage, such as mortal-ity, fecundity, and dispersal rates.

Despite these underlying differences, the two model-ing approaches produced similar patch occupancy rates



for lizards in Florida scrub. Moreover, both models pre-dicted field measurements of patch occupancy with rea-sonable accuracy in south and north parts of the scrublandscape. Both models also accurately predicted whichpatches were occupied. But model agreement may havebeen enhanced by the prevalence of small patches,which were mostly unoccupied, and large patches (�12ha), which were nearly always occupied ( Fig. 3). Thisoccupancy pattern is typical of mainland-island systems,which consist of large patches that act as extinction-immune mainlands and small patches that act as extinc-tion-prone islands ( Boorman & Levitt 1973; Harrison1991; Stith et al. 1996). The dynamics of mainland-islandsystems are relatively simple. Also, because we parame-terized our models using data from Avon Park, the accu-racy and agreement of the two models may be reducedfor other landscapes with more complex dynamics.

Model Comparison in Artificial Landscapes

The wide range of patch sizes and interpatch distanceswe modeled on artificial landscapes produced popula-tion dynamics characteristic of different types of meta-populations (sensu Harrison 1991). Some patch configu-rations that exhibited partial occupancy at quasi-equilibrium(Fig. 5) represent “classical” metapopulations consisting ofsmall patches that go extinct and are recolonized. Wealso observed “nonequilibrium” metapopulations, in whichrapid extinction of patches led to extinction of the entiremetapopulation (Harrison 1991; Stith et al. 1996), and land-scapes with patches that never went extinct. These differ-ing population dynamics provided substantial opportunityto produce differences between the incidence functionmodel and the stage-based matrix model. The modelsshowed relatively high agreement for most landscapeconfigurations but produced very different results intwo regions, small-near patches and large-far patches.

In the small-near patch configuration, the stage-basedmatrix model predicted low metapopulation persis-tence, whereas the incidence function model predictedhigh persistence. Our congruency analysis indicated thatthe incidence function model would predict low persis-

Figure 5. Results of transient dynamic simulations for artificial landscapes with 20 patches for (a) the stage-based matrix model and ( b) the incidence function model. The upper limits of the contour shading in (a) and ( b) represent 20%, 40%, 60%, and 80% occu-pancy, respectively. Part (c) shows the difference be-tween the matrix model and the incidence function model. For (c), the darkest shading represents differ-ences of �50%, the lightest shading represents differ-ences between 20% and 50%, and moderate shading represents differences of �20%.



Table 4. Analysis of the congruence of the stage-based matrix model and incidence function model on artificial landscapes of 20 patches.

Model

Patchsize(ha)

Interpatchdistance

(m) ParameteraRange of

manipulation Congruenceb Valuec

Stage-basedd matrix 2 50 p 0.1–1.0 no —� 0.0–0.016 no —

fecundity 6.16–22 yes 22e

stage 1 survivorship 0.12–0.22 yes 0.22e

stage 2–3 survivorship 0.06–0.22 no —fecundity stage 1 survival 6.16–9.9 0.12–0.16 yes 9.9 0.16

Incidence functiond 2 50 � 0.076–10.0 no —x 4.0–0.974 no —� 0.016–0.07 yes 0.062e

y� 4.04–100 yes 43e

� y� 0.016–0.03 4.04–32 yes 0.03 22e

Stage-based matrix f 32 750 p 0.0–0.1 no —� — — —

fecundity 5.5–11 yes 6e

stage 1 survivorship 0.11–0.18 yes 0.11e

stage 2–3 survivorship 0.0–0.11 no —fecundity stage 1 survival 8–11 0.14–0.18 yes 8 0.14e

Incidence function f 32 750 � 0.0–0.076 yes 0.0008e

x 0.974–100 yes 100e

� 0.0–0.016 yes 0.008e

y� 0–4.04 yes 0.01e

� x 0.001–0.076 0.974–3.0 yes 0.02 2.0aAll parameter values are as in the transient dynamics simulations except for the parameter(s) being manipulated. Parameter refers to the pa-rameter(s) being manipulated. The parameter p is the percentage of stage 1 migrants leaving a patch.bIndicates whether the respective model’s output came within 5% of the alternative model’s output for the percentage of occupied patches.cValue of the parameter(s) for which congruence was achieved.dIn this landscape the stage-based matrix model predicts 0% patch occupancy, and the incidence function model predicts 99% occupancy ( Fig. 5).eSuggests a value that is unrealistic or inconsistent with observations for scrub lizards.fIn this landscape the stage-based matrix model predicts 100% occupancy, and the incidence function model predicts 1.8% occupancy.

tence only by reducing patch turnover and dispersalability to levels much lower than those we observed inthe field. In contrast, the matrix model could be broughtinto congruence with the incidence function model byincreasing fecundity and survivorship of stage 1 juve-niles to levels that exceed those measured for smallpatches in the field but fall slightly below the field val-ues for large patches. In artificial landscapes with smallpatches, each patch produced few dispersers that weredistributed equally among 19 equally distant patches.Thus, even when dispersal rates were high, the totalnumber of immigrants into each patch was low. In amore realistic landscape, a smaller number of nearbypatches would receive the majority of dispersers, thusincreasing the “rescue effect” ( Brown & Kodric-Brown1977 ) and metapopulation persistence. This may ex-plain why increases in the dispersal parameters couldnot produce persistent metapopulations in the stage-based matrix model for the small-near configuration.

In the large-isolated patch configuration, the inci-dence function model predicted that larger patchesnearly always went extinct. By simultaneously decreas-ing � (probability that a patch of unit size will go ex-tinct) and increasing x (coefficient relating patch sizeand extinction probability) to values that remained bio-

logically reasonable, the predictions of the incidence func-tion model became compatible with the stage-based matrixmodel. In contrast, no plausible parameter manipulationsof the stage-based matrix model resulted in predictionscompatible with the original predictions of the incidencefunction model. Although not definitive, these results sug-gest that the incidence function model may underesti-mate the persistence of larger populations in this land-scape configuration.

Advantages and Disadvantages of the TwoModeling Approaches

A major advantage of incidence function models overstage-based matrix models is the relatively simple fielddata requirements, which can be met from a single com-prehensive patch survey of presence-absence. There are,however, some important restrictions on the type of sys-tem for which an incidence function model is appropri-ate. The study area must be completely surveyed, or theinfluence of unsurveyed patches may lead to overesti-mates of migration ability. Hanski (1999) recommendsthat surveyed patches constitute a small fraction of thetotal landscape (e.g., �20%) and that 10 or more patchesshould be occupied and 10 or more should be unoccu-



pied. Furthermore, there should be a sufficient numberof patches (at least 30) with enough variation in patchsize and isolation to provide good parameter estimates.Most important, the system should be in a state of “sto-chastic extinction-colonization quasi-equilibrium” (Hanski1999). The incidence function model assumes that thelandscape and pattern of patch occupancy have beenfairly stable or, if the landscape has been recently altered,that sufficient time has passed for the populations toreach an equilibrium state within the altered landscape.

In recently restored landscapes, patch occupancy maybe below the equilibrium level due to a lag in coloniza-tion. But the opposite situation is much more common:landscapes that have experienced recent destruction ordegradation may show higher occupancy rates than theequilibrium level due to a “crowding effect” (Lambersonet al. 1992), or temporary persistence of a few long-livedindividuals. The incidence function approach also doesnot distinguish between sources and sinks. The pres-ence of a species in sink patches may depend com-pletely on immigration from a source patch (Pulliam1988), which may recently have been destroyed. Thus,presence-absence surveys may seriously overestimatethe persistence of many species that have not yet “re-laxed” into an equilibrium state with much lower patchoccupancies. Therefore, careful consideration should begiven to the recent history of a study area.

Although stage-based matrix models have greater de-mands for field data than incidence function models,these models have the advantage in that they do not as-sume that a system is in a state of quasi-equilibrium. Theoccupancy of a patch is determined not simply by itssize or degree of isolation, but also by the survival, fe-cundity, migration rates, and carrying capacities estimatedfor each patch. These demographic parameters can be var-ied from patch to patch, reflecting source-sink dynamicsor differences in carrying capacity. The more mechanisticapproach of explicitly modeling local population dynam-ics gives stage-based matrix models an advantage over in-cidence function models in the common situation wheresystems are not at equilibrium. Estimating demographicrates and other parameters used by matrix models greatlyincreases the demand for field data, however, and in-creased model complexity also brings greater risk of errorpropagation due to poor parameter estimation (Conroy etal. 1995; Beissinger & Westphal 1998; Hanski 1999).

Our results for the scrub landscape suggest that the in-cidence function approach may provide a relatively eco-nomical first step in the landscape modeling process.Within our study area, the incidence function modelwas much easier to parameterize than the stage-basedmatrix model because there was no need to estimate vi-tal rates for lizard populations in small and large patchesor to estimate local population abundance. Also, therewas no need to directly estimate migration rates or tomake limiting assumptions about the direction of dis-

persal because the patch-specific colonization probabil-ity was a function of patch isolation. With additional data(e.g., assessment of long-term stochastic effects on pop-ulations, correlation of demographic processes amongpatches, and dispersal rates), the stage-based demographicmodel presumably would be a more realistic and flexiblemodel than the incidence function model. Nevertheless,the incidence function model performed as well as thematrix model in predicting patch occupancy and re-gional population dynamics in our Florida scrub land-scape, and both models gave similar predictions whenused to examine different management scenarios.

Our study of artificial landscapes suggests that thesetwo models may differ substantially for some landscapeconfigurations, highlighting an issue that merits furtherstudy. When appropriate data are available, we suggestthat researchers use both techniques and examine differ-ences in model output. Coexisting species may respondquite differently to the same spatial structure within agiven landscape (e.g., Hokit et al. 1999). As conservationbiologists attempt to incorporate spatial structure intothe assessment of population viability, reliable land-scape-level population models will be needed. The feasi-bility of obtaining data for population modeling varieswidely among species and limits modeling approachesappropriate for each species. Understanding the limita-tions and applicability of different modeling approacheswill become increasingly important as conservation pro-grams undertake the challenge of managing for multiplespecies in a landscape context.

Acknowledgments

A large number of people at the Avon Park Air ForceRange provided support for this project. We particularlythank P. Walsh, B. Progulske, and P. Ebersbach. Fieldwork was accomplished with great dedication from thefollowing assistants: J. Matter, J. Arnett, K. Griffis, K. Ko-sel, K. Long, P. Mikula, and C. O’Brien. Funding for thisproject was provided, in part, by the U.S. Department ofDefense in collaboration with the Florida CooperativeFish and Wildlife Research Unit of the National Biologi-cal Service, the Species at Risk Initiative of the NationalBiological Service, and the Florida Game and FreshwaterFish Commission. This is Florida Agricultural Experi-ment Station Series R-07755.

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