a habitat-based metapopulation model of the...

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Conservation Biology, Pages 422–434Volume 11, No. 2, April 1997

A Habitat-Based Metapopulation Model of the California Gnatcatcher

H. RE T AKÇAKAYA* AND JONATHAN L. ATWOOD†

*Applied Biomathematics, 100 North Country Road, Setauket, NY 11733, U.S.A., email [email protected]†Manomet Observatory for Conservation Sciences, P.O. Box 1770, Manomet, MA 02345, U.S.A.

Abstract:

We present an analysis of the metapopulation dynamics of the federally threatened coastal Califor-nia Gnatcatcher (

Polioptila c. californica

) for an approximately 850 km

2

region of Orange County, California.We developed and validated a habitat suitability model for this species using data on topography, vegetation,and locations of gnatcatcher pair observations. Using this habitat model, we calculated the spatial structureof the metapopulation, including size and location of habitat patches and the distances among them. We useddata based on field studies to estimate parameters such as survival, fecundity, dispersal, and catastrophes,and combined these parameters with the spatial structure to build a stage-structured, stochastic, spatially-explicit metapopulation model. The model predicted a fast decline and high risk of population extinction withmost combinations of parameters. Results were most sensitive to density-dependent effects, the probability ofweather-related catastrophes, adult survival, and adult fecundity. Based on data used in the model, the great-est difference in results was given when the simulation’s time horizon was only a few decades, suggesting thatmodeling based on longer or shorter time horizons may underestimate the effects of alternative managementactions.

Modelo de Metapoblación Basado en el Hábitat de

Polioptila c. Californica

Resumen:

Presentamos un análisis de la dinámica metapoblacional de


, especieamenazada de extinción de las costas de California, para una región de aproximadamente 850 km

2

delcondado de Orange, California. Describimos y validamos un modelo de ajuste para esta especie utilizandodatos de topografia, vegetación y sitios de observación de la especia. Utilizando este modelo de hábitat, esti-mamos la estructura espacial de la metapoblación incluyendo tamaño y localización de parches de hábitatasí como las distancias entre ellos. Utilizamos datos basados en estudios de campo para estimar parámetrostales como la supervivencia, fecundidad, dispersión y catastrofes, también se combinaron estos parámetroscon la estructura espacial con la finalidad de construir un modelo metapoblacional estructurado por pasos,estocástico y espaciálmente explícito. El modelo predijo una declinación rápida y un alto riego de extinciónde la población con la combinación de la mayoría de los parámetros. Los resultados fueron mas sensitivos aefectos densodependientes, la probabilidad de catastrofes ocasionadas por el clima, la supervivencia de adul-tos y la fecundidad de los adultos. Basados en los datos utilizados en este modelo, las diferencias mayores enlos resultados se produjeron cuando el horizonte de tiempo de las simulaciones fue de solo unas cuantasdécadas, sugiriendo que el modelado basado en horizontes de tiempo largos o cortos pueden desestimar los

efectos de acciones alternativas de manejo.

S I

Introduction

The California Gnatcatcher’s northernmost subspecies(


) has declined due to extensive

agricultural and urban development of coastal sage scrub,the species’ primary habitat type in southern Californiaand northwestern Baja California (Atwood 1993). Listedin 1993 as threatened under the U.S. Endangered SpeciesAct (ESA), protection of the gnatcatcher and its habitathas become a major focus in the inaugural application ofthe State of California’s Natural Community Conserva-

Paper submitted May 15, 1996; revised manuscript accepted October10, 1996.

Conservation BiologyVolume 11, No. 2, April 1997

Akçakaya & Atwood California Gnatcatcher Metapopulation

423

tion Planning (NCCP) program (Atwood & Noss 1994).This voluntary, regional land-use planning process,aimed at delineating core tracts of essential natural habi-tat while allowing economic development in areas oflower biological significance, has been described as apotential model for resolving conflicts between conser-vation and economic concerns (Reid & Murphy 1995).Because participation in the NCCP program was for-mally recognized as an alternative to the requirementsfor incidental “take” permits as authorized under Section10(a) of the ESA, most NCCP efforts in southern Califor-nia have revolved around planning decisions intended toconserve gnatcatchers and their habitat.

We analyzed the viability of a California Gnatcatchermetapopulation in central and coastal Orange County.Our aims were (1) to develop a habitat suitability modelfor California Gnatcatcher, (2) to demonstrate how thismodel can be linked to a metapopulation model for via-bility analysis, and (3) to analyze the sensitivity of theviability of this species to uncertainties in model param-eters.

The NCCP plans that were recently prepared for cen-tral and coastal Orange County did not incorporatethese results, in part because our analysis was not com-pleted until after the planning process was well under-way. Nonetheless, it is our hope that these results willcontribute to future NCCP planning efforts elsewherein southern California, as well as provide a basis forlong-term habitat management within the NCCP coastalsage scrub reserve system in central and coastal OrangeCounty.

The Model

We developed a spatially-explicit, stage-structured, sto-chastic model of the California Gnatcatcher metapopula-tion based on habitat suitability maps we developed andon demographic data from Atwood et al. (1995, unpub-lished manuscript). We used the population viability anal-ysis program RAMAS/

GIS

, which is designed to link land-scape data from a geographic information system with ametapopulation model (Akçakaya 1995; for another ap-plication of the program see Akçakaya et al. 1995; for re-views see Kingston 1995 and Boyce 1996).

We used data on the current distribution of the spe-cies’ habitat to find the spatial structure of the metapop-ulation (i.e., to identify the location, size, and shape ofhabitat patches in which [sub]populations of the meta-population exist). In addition to spatial structure, we in-corporated parameters related to demography, such ascarrying capacities, initial abundances and vital rates ofeach population, the amount of year-to-year variability invital rates, as well as the rate of dispersal betweenpatches and the degree of similarity of environmentalfluctuations that different populations experience.

We accounted for uncertainties arising from measure-ment errors and lack of data by making three estimatesof each parameter (i.e., estimating lower and upperbounds, in addition to a “medium” estimate). We usedthese ranges to estimate upper and lower bounds on theestimated viability of the species. In most cases the esti-mation of a range is quite arbitrary; we used ranges suchas

6

20% to 100% for parameters based on crude esti-mates and ranges such as

6

5% to 10% for parametersbased on more reliable data.

Habitat and Census Data

The habitat variables and census data that formed the ba-sis of our analysis were provided by the GIAS lab ofSouthern California Edison in the form of digital raster(grid) maps with a resolution of 100 m. These maps de-scribed topography, vegetation, and gnatcatcher distri-bution in approximately 853 km

2

of central and coastalOrange County.

Topographical data layers included elevation, slope,and aspect, with each grid cell being represented by thevalue at the cell’s central point. The values for elevationmap (named ELV) are elevation in meters above sealevel, the values for the slope map (SLP) are in percent-ages (i.e., 0 for flat areas; 100 for a 45

8

incline), and thevalues for aspect (ASP) are in units of degrees fromnorth (e.g., 180 for south, 90 for east and west, 0 fornorth), ranging from 0 to 180.

Vegetation maps, originally prepared by Jones andStokes (1993) from color aerial photographs, were con-verted from ARC/Info coverages to grid (raster) format.Because field experience suggested that the Jones andStokes (1993) classification of various coastal sage scrubsubassociations might include a relatively high degree ofsubjective interpretation and because a preliminary anal-ysis failed to detect any consistent patterns regardingpresence or absence of gnatcatchers in different sub-associations, we combined the original 20 subassociationsof coastal sage scrub into a single, generalized map. Thetotal amount of coastal sage scrub was calculated foreach map cell, with values ranging from 0 m

2

(no coastalsage scrub) to 10,000 m

2

(full cover). Because of uncer-tainties regarding accuracy of the raw vegetation data,each cell in the coastal sage scrub (CSS) data layer wasassigned the value of a 9-cell (3

3

3) moving average ofthese area estimates.

We also used the Jones and Stokes (1993) vegetationdata to create maps of wetland vegetation (including ri-parian habitat), woody vegetation (combining chaparral,woodland, and forest categories), and grasslands. Basedon these maps, we created three data layers describingdistance of each cell to (1) the nearest cell with at least10% cover of trees or other woody vegetation (DTR), (2)the nearest cell of grassland (DGR), and (3) the nearest

424

California Gnatcatcher Metapopulation Akçakaya & Atwood


cell of wetland or riparian vegetation (DWT). We be-lieve that variables describing distance to woody vegeta-tion or grassland provide a more objective characteriza-tion of coastal sage scrub subassociations than do theclassifications used by Jones and Stokes (1993). For ex-ample, coastal sage scrub dominated by

Artemisia cali-fornica

generally occurs in closer proximity to grasslandareas, whereas subassociations dominated by

Salviamellifera

often occurs near chaparral or oak woodland(and distant from grassland). We truncated distance val-ues greater than 2 km for these data layers.

In surveys of California gnatcatchers (conducted inOrange County in 1991–1992 and 1994), 325 pairs and80 single individuals were found in 1991–1992, and 238pairs and 80 single individuals were located in 1994.Cells where one or more pairs were recorded were as-signed a value of 1 (present) (

n

5

547). In a separatedata layer, which we used for validation purposes, wealso assigned a value of 1 (present) to all cells (

n

5

129)where single birds were observed.

Finally, because distributional surveys did not cer-tainly document gnatcatcher absence within the studyarea, we randomly added 273 (50% of the number ofcells where pairs were observed) points intended to rep-resent locations unsuitable for gnatcatchers. We se-lected these “absent” localities randomly, but with thefollowing constraints. (1) “Absent” localities were notplaced within 300 m of cells where pairs or single indi-viduals were observed. Because the species’ territorysize may be larger than one 1 ha (Atwood et al., unpub-lished manuscript), it is likely that cells surrounding anobservation point would support suitable habitat eventhough no observations were obtained in these cells.Similarly, we consider it likely that many observations ofsingle birds occurred in suitable breeding habitat andthat areas near these points should not be used to char-acterize unsuitable habitat. (2) Absent localities werenot placed within 300 m of other randomly-placedpoints because too much clustering would lead to re-dundancy and not add information to the analysis. (3)Absent localities were not placed in cells dominated byagricultural fields, water, rural development, or residen-tial development. All of these habitat types are clearlynot suitable gnatcatcher habitat, and their inclusionwould not add information to the analysis.

Habitat Suitability Map and Patch Structure

We used logistic regression (SAS Institute 1990) to calcu-late a habitat suitability function, which was then usedto calculate an index of habitat suitability for each cell.Logistic regression is indicated in cases where the re-sponse (dependent) variable is binary (e.g., 0 or 1). Weused a stepwise approach with a significance level of

p

5

0.05 for adding and removing variables. After all vari-ables were tested, we started adding interaction terms

(and removing those that became non-significant). Inthis phase we retained all previously-added single vari-ables, even if they were rendered not significant withthe addition of an interaction term.

The link between the habitat map and the metapopu-lation model was characterized by two parameters.

Thresh-old HS

is the minimum habitat suitability (HS) value be-low which the habitat is not suitable for reproductionand/or survival.

Neighborhood distance

is used to iden-tify nearby cells that belong to the same patch and mayrepresent, for example, the foraging distance of the spe-cies. These parameters are used by a patch-recognitionalgorithm that delineates patches on the habitat map.Given these two parameters, the program finds clusters orgroups of nearby cells (i.e., within the neighborhood dis-tance of each other) that have HS values higher than orequal to the threshold HS and labels them as patches.

We used 0.5 (i.e., 50%) as the threshold HS; only thosecells that had a habitat value of 0.5 or above were con-sidered when habitat patches were analyzed. We used aneighborhood distance of 3 cells, which corresponds tothe assumption that any two suitable locations withinabout 300 m of each other are in the same habitat patch.We analyzed the sensitivity of extinction risk to thesetwo parameters by using a range of 0.45 to 0.55 for thethreshold HS parameter and a range of 250 m to 350 mfor the neighborhood distance parameter (Table 1).

Carrying Capacities and Initial Abundances

The program allows the calculation of carrying capaci-ties (

K

) based on the total habitat value of each patch(i.e., the sum of habitat values of all cells that are in-cluded in a patch). We estimated carrying capacitiesbased on territory sizes, which were estimated to be4.69 ha on average in Orange County (Bontrager 1991).We used 1/4.69 (

5

0.213) as a scaling constant in calculat-ing

K

of each patch by multiplying it with the total habi-tat value in each patch. We used total habitat value in-stead of the total area as the argument because the latterincludes areas with low habitat value. The sum of habi-tat values in a patch is also related to the area of thepatch, but weighted by the amount of habitat in eachcell, so that territories in areas with low habitat value areassumed to be larger. We used

6

20% of this parameteras the upper and lower limits. We excluded patcheswith

K

#

5 because only larger populations would haveat least one pair of adults at stable stage distribution.

We specified the initial number of individuals in eachpatch as a fixed proportion of the carrying capacity of thatpatch. We estimated this proportion based on the ratio ofthe total number of observed pair locations (for both 1992and 1994, but excluding the 1994 locations within 150 mof 1992 locations) (503) to the number of pairs (629) pre-dicted by the above calculation of the total carrying capac-ity. Thus, each patch had a population of 503/629, or 80%


Akçakaya & Atwood California Gnatcatcher Metapopulation

425

of its carrying capacity at the beginning of the simulation.For the lower limit we used 52% (based on only the 1992pairs), and for the upper limit we used 100% (assumingsome of the singles represent pairs). For all simulations,we assumed the populations have a stable distribution ofindividuals to stages at the start of the simulation.

Stage Structure

We modeled the dynamics within each patch with astage-structured, stochastic matrix model with two stages(juveniles and adults). In parameterizing this stage-struc-tured model, we assumed (1) all reproduction in thepopulation takes place in a relatively short breeding sea-

son (a “birth-pulse” population, Caswell 1989); (2) thepopulation is censused immediately after each breedingseason (a post-reproductive census, Caswell 1989); (3)all adults breed (so that the proportion of last year’sadults who are breeders this year is simply the survivalrate of adults); (4) the maternity rate (number of fledg-lings per breeder) is the same whether none, one, orboth of the breeders are nesting for the first time (i.e.,were juveniles last year); and (5) the stage matrix is thesame in all populations. With these assumptions, thestage matrix is

,

where

S

a

is survival rate of adults;

S

j

is survival rate of ju-veniles;

P

JB

is proportion of last year’s juveniles that arebreeders this year; and

M

is maternity or fertility (num-ber of fledglings per breeder).

The two elements in the first row of the matrix are fe-cundities. Adult fecundity (

F

a

) is equal to

S

a

·

M

, and juve-nile fecundity (

F

j

) is equal to

P

JB

·

M

. We estimated

P

JB

,

S

a

,and

S

j

based on data from the California Gnatcatcher popu-lation on the Palos Verdes peninsula (Atwood et al., unpub-lished manuscript), and

M

based on data from OrangeCounty (Woehler, unpublished manuscript; Bontrager etal., 1995). There were no comparable data on survivalfrom Orange County. The data set represented threetransitions: 1993–1994, 1994–1995, and 1995–1996.

The data for 1994–1995 suggests a sharp decline,which is what was observed on the Palos Verdes Penin-sula (Atwood et al., unpublished manuscript), as well asin coastal Orange County (Erickson & Pluff, unpublishedmanuscript; Chambers Group and LSA Associates, unpub-lished data). This may be a result of the exceptionally wetand cold winter of 1994–1995. One hypothesis is thatgnatcatchers, because of their small body size and highmetabolic rates, have higher mortality under such weatherconditions (e.g., Mock, unpublished manuscript). If thisis correct, the survival rates for 1994–1995 would be anexception rather than the rule. For this reason, we usedthe average of 1993–1994 and 1995–1996 transitions toestimate the average stage matrix and the 1994–1995data as a basis for modeling catastrophes (Table 2).

To calculate upper and lower bound on vital rates, weassumed a measurement error of

6

1 individual in the re-covery (re-sighting) of banded individuals. The rationalefor these limits is based on the assumption of birth-pulsepopulation. The parameters are estimated assuming thatthe census is made exactly at the same time each year. Ifthe census is actually made, say, one day earlier than thisexact time, then the re-sighting might be an overesti-mate (if a bird actually dies before the “correct” time).At a practical level,

6

a number for re-sighting is morereasonable than, say,

6

10% of the vital rate itself be-cause the latter does not account for differences in sam-

PJB M⋅

Sj

Sa M⋅

Sa

Table 1. Low, medium, and high estimates of parameters used in the model of the California Gnatcatcher metapopulation.

Parameter estimate

Parameter Low Medium High

Habitat-demographylink

Threshold HS 0.45 0.50 0.55Neighborhood

distance (m)250 300 350

Carrying capacity(

K

) multiplier

a

0.170 0.213 0.256

Initial abundance(as % of

K

)52 80 100

Mean dispersaldistance (km)

2.5 3.0 3.5

Stage matrixJuvenile survival

rate (

S

j

)0.3275 0.3441 0.3606

Juvenile fecundity(

F

j

)0.4901 0.5376 0.5874

Adult survival rate(

S

a

)0.4650 0.4975 0.5300

Adult fecundity(

F

a

)0.7975 0.8899 0.9876

Density dependenceDensity dependence

typeceiling

c

contest

c

Allee effects (localthreshold as % of K)

0 2 4

Density-dependentdispersal

none weak strong

StochasticityStandard deviations

b

0.8

S S

1.2

S

Correlation offluctuations (

b

)10 30 100

Catastrophe (weather)probability

0.07 0.14 0.28

Number of fires 1 2 4Metapopulation

extinction threshold 30 60 120

a

This parameter is multiplied with the total habitat suitability value of the patch to calculate carrying capacity and initial abundance in the patch.

b

S

represents the standard deviations given in Table 2.

c

See text, page 426, for details.

426

California Gnatcatcher Metapopulation Akçakaya & Atwood


ple size for different vital rates. The number 1 (instead of2) is used as a minimum, but even this gave quite largeuncertainties at the risk level. The resulting ranges arelisted in Table 1.

Environmental and Demographic Stochasticity

We modeled environmental stochasticity by samplingthe set of vital rates used to project the dynamics of eachpopulation from random (lognormal) distributions withmeans taken from the mean stage matrix and standarddeviations taken from a “standard deviations matrix.”The sampling was done at each time step (in this caseeach year), thus the required standard deviation is oneamong years. We estimated these standard deviationsbased on the variability of survival rates and fecunditiesfrom the two transitions (1993–1994 and 1995–1996;Table 2). We used the 1994–1995 transition to model ca-tastrophes, so we did not include this transition in esti-mating the standard deviations.

We incorporated demographic stochasticity by sam-pling number of survivors from a binomial distributionand number of offspring from a Poisson distribution(Akçakaya 1991). In addition, we incorporated demo-graphic stochasticity in dispersal.

Catastrophes

One type of catastrophe that may affect species living incoastal sage scrub is fire. Coastal sage scrub is frequentlysubject to fire, especially in areas where the habitat oc-curs in close proximity to human population centers(Westman 1982; Keeley 1982). Two fires have occurredwithin the study area since 1948 (Fig. 5 in NCCP HabitatConservation Plan prepared by R. J. Meade Consulting,Inc. 1995). In 1982 a fire was located in the area identi-fied here as the second largest patch, and in 1993 oneoccurred in the area identified as the largest patch. The1993 fire destroyed habitat occupied by approximately127 gnatcatcher pairs (Atwood et al., unpublished manu-script), corresponding to about 48% of the initial abun-

dance in the modeled population occupying the largestpatch. Based on this we assumed two fires, one affectingthe second largest patch in year 34 and the other affect-ing the largest patch in year 45 in the 50-year simula-tions of our model. We assumed that each fire will de-crease

K

by 48% and

K

will increase to its original levelin 10 years in each case. Although these assumptions ig-nore long-term affects of fires, such as potential conver-sion of coastal sage scrub to grassland (Anderson 1991),we believe they capture most of its short and medium-term effects on gnatcatcher populations. We analyzedthe sensitivity of results to fire by running simulationswith only one fire (in the largest patch), and with twoadditional fires affecting the largest and second largestpatches in years 25 and 14, respectively.

Another type of catastrophe with direct impact on gnat-catcher populations may be extreme weather conditions,such as those that may have characterized the winter of1994–1995. In this model, we used the demographic pa-rameters estimated from 1994–1995 to model such catas-trophes. We assumed that the effect of such a catastropheis a sharp decline in vital rates beyond the normal year-to-year fluctuations (see

Environmental and DemographicStochasticity

, above). After a catastrophe year, the vitalrates are again sampled from the average stage matrix (es-timated based on the average of 1993–1994 and 1995–1996transitions) with the standard deviations given above.

The frequency of such extreme population fluctu-ations caused by weather conditions is difficult, if not im-possible, to estimate because it is not clear which aspectof the weather makes the crucial difference. A statisticalestimation is possible only after several years of vital rateestimates. In the absence of such information, we focusedon the minimum temperatures in October through De-cember. In the two weather stations closest to the studyarea (Tustin and Newport), frequency of average of thethree monthly minimum temperatures less than or equalto that in 1994 was about 14% in the last 98 years. Thuswe chose the annual catastrophe probability of 0.14. Weused the range 0.07–0.28 to evaluate the sensitivity of re-sults to this assumption.

Density Dependence and Allee Effects

For most simulations we assumed a ceiling-type densitydependence model for each population and used thecarrying capacities calculated based on habitat data aspopulation ceilings. This model allows the populationsto fluctuate independent of the population size (

N

), ac-cording to the stage matrix and the standard deviationsmatrix, until the population reaches the ceiling. Thepopulation then remains at this level until a populationfluctuation takes it below the ceiling. We also modeleddensity dependence with a contest model (which usesthe Beverton-Holt equation) with the assumption that

Table 2. Stage matrix parameters: juvenile survival rate (

S

j

), proportion juveniles that become breeders (

P

JB

), juvenile fecundity (

F

j

5

P

JB

·

M

), adult survival rate (

S

a

), and adult fecundity(

F

a

5

S

a

·

M

).

Year

M S

j

P

JB Fj Sa Fa

1993 1.7a 0.2568c 0.1757c 0.2986 0.5200c 0.88401994 2.3a 0.1558c 0.0909c 0.2102 0.1892c 0.43751995 1.9b 0.4314c 0.4118c 0.7766 0.4750c 0.8958

Average for (93,95) 0.3441 0.2937 0.5376 0.4975 0.8899SD for (93,95) 0.0873 0.1180 0.2390 0.0225 0.0059aWoehler (unpublished manuscript)bAverage of M from Woehler (unpublished manuscript) and Bon-trager, et al. (1995)cAtwood et al. (1995, unpublished manuscript)


Akçakaya & Atwood California Gnatcatcher Metapopulation 427

each population grows with a rate of 15% when N is lowand is stable with 0% growth when N 5 K.

Allee effects, which may cause a reduction in vitalrates when population size becomes very small, are notwell-studied for this species. In the current model we in-corporated Allee effects by specifying a local extinctionthreshold for each population. Once any population fallsto or below its local threshold, the model assumes thepopulation to be extinct by setting the abundance tozero. The patch then remains unoccupied, unless it iscolonized by dispersers from another patch. We set thelocal thresholds at 0%, 2%, and 4% of the carrying capac-ity of the patch. By considering a population to be ex-tinct once it reaches or falls below its threshold, themodel need not accurately predict the dynamics of thepopulations at these low abundance levels.

In addition, we specified a metapopulation extinctionthreshold of 30, 60, or 120 individuals (about 1%, 2%, and4% of the total initial metapopulation size) and calculatedthe viability results in terms of falling below this threshold.

Metapopulation Dynamics: Correlation-distanceFunction and Dispersal

Parameters related to dynamics at the metapopulationlevel include the interdependence of environmentalfluctuations among populations and patterns of dis-persal. The relatively small study area (with the maxi-mum distance between any two patches about 40 km)and the apparent dependence of gnatcatcher biology toweather conditions suggests that weather patterns suchthose seen in southern California (affecting substantialareas) impose a correlation structure on the metapopula-tion dynamics. In a study of the southern Californiametapopulation of the Spotted Owl, LaHaye et al. (1994)found strong, distance-dependent spatial autocorrelationamong rainfall and used this relationship between rain-fall correlation and distance as a basis for setting correla-tion among population dynamics. In our model we usedthree different correlation-distance functions to set thecorrelation of vital rates among populations (Fig.1). Thefunction is C 5 exp(2d/b), where C is the coefficient ofcorrelation between the vital rates of two populations, dis the distance (km) between the centers of these twopopulations, and b is a parameter that describes how fastthe correlation declines with increased distance betweenpopulations. We used b values of 100 (high correlation),30 (medium correlation), and 10 (low correlation).

In our model dispersal refers to the movement of birdsamong habitat patches, and dispersal rate (proportiondispersing from target population to source population)may depend on the distance between source and targetpopulations, the abundance in the population, andwhether the bird is a juvenile or an adult.

We estimated the distance dependence of dispersalbased on data for 1993 and 1994 from Atwood et al. (un-

published manuscript), who report the number of juve-niles that dispersed different distances. We divided thenumber of dispersing juvenile birds in each distanceclass by the total sample size (32) to obtain proportiondispersing, and used this as the dependent variable. Weused the mid-point of each distance class (in km) as theindependent variable (d) in the exponential model M 5a ? exp(2d/b). In our model M is the dispersal rate, d isthe distance (km), and a and b are model parameters.The parameter b is the average dispersal distance. Themodel was fitted with a 5 0.4 and b 5 2.5 km (Fig. 2).In addition, the above equation was modified to reflect amaximum dispersal distance of 15 km. Because of therelative isolation and small size of the Palos Verdes pen-insula where these data were collected, we used the fit-ted function to model minimum dispersal, and set theaverage dispersal distance to 3.0 and 3.5 km for mediumand high dispersal. The distance metric we used in thisstudy is one between the center of the source popula-tion to the edge of the target population. The asymmetryof this measure of distance allows for more realisticmodeling of dispersal between a large and a small patch.

We assumed adults have negligible dispersal amongpopulations (they can disperse within the same patch)and used dispersal parameters discussed above only forthe juvenile stage. We incorporated demographic sto-chasticity in dispersal among populations by samplingthe number of dispersers from a binomial distributionwith sample size equal to the number of juveniles in thesource population and probability equal to the dispersalrate based on distance.

We modeled density dependence in dispersal for eachpopulation such that the dispersal rate was directly pro-portional to population size. We modeled two levels of

Figure 1. Correlation-distance functions [y 5 exp(2x /b)] used in the model. The function gives the correla-tion between the vital rates of two populations sepa-rated by the indicated distance.

428 California Gnatcatcher Metapopulation Akçakaya & Atwood


density dependence in dispersal, in addition to density-independent dispersal (Fig. 3). Under strong density-dependent dispersal, when the population size (N) islower than the carrying capacity (K), the proportion dis-persing is lower in proportion to the ratio of N/K. Whendispersal is density-independent (“none” on the graph),the dispersal rate is M regardless of abundance.

Analysis and Viability Measures Used

The analysis of the dynamics of the California Gnat-catcher metapopulation with the model describedabove consisted of a series of simulations. Each simula-tion consisted of 10,000 replications, and each replica-tion projected the abundance of the each population for50 time steps (years). At each time step the number ofjuveniles and adults at each population were projectedusing a set of vital rates drawn from a random (lognor-mal) distribution.

The mean values of these vital rates (survival and fe-cundity) were taken from the stage matrix, and the stan-dard deviation of vital rates are those given by the stan-dard deviation matrix. The sampled stage matrices ofdifferent populations were correlated according to one ofthree sets of correlations, whereas the elements of thestage matrix within a population were perfectly corre-lated. The number of individuals in each stage of eachpopulation, as well as the number of dispersers were al-ways integer numbers. These calculations followed thealgorithm described by Akçakaya (1995).

To analyze the sensitivity of model results to each pa-rameter, we ran three simulations for each parameter,using the lower, medium, and upper estimates of that

parameter and the medium estimates of all the other pa-rameters (Table 1). We used three measures to expressthe predicted viability of the metapopulation: (1) me-dian time to fall below the metapopulation extinctionthreshold, (2) risk of falling below the metapopulationthreshold anytime within 20 years, and (3) risk of fallingbelow the threshold anytime within 50 years.

Results

Habitat Suitability Map

The results of the logistic regression analysis are summa-rized in Table 3. The goodness of fit statistics given at theend of the table show that the regression is highly signifi-cant, and the probability column shows that regressioncoefficients are also statistically significant. Slope, as-pect, and distance from wetlands were non-significant.

The frequency distribution of habitat suitability valuesin the landscape is given in Fig. 4. Only 24% has habitatvalues of 0.5 or above, and the average habitat suitabilityis about 0.30. The locations where gnatcatcher pairswere observed have habitat values ranging from 0.14 to1.00, with an average of 0.86. The frequency distribu-tion of the habitat suitability values of locations withgnatcatcher observations is given in Fig. 5a. About 95%of the observations are in locations with habitat valuesof 0.5 or above.

Figure 3. Total rate of dispersal as a function of popu-lation size, used to model density-dependent dispersal for each population. M is the total rate of dispersal from the population to all other populations, given by the dispersal distance function. K is the carrying ca-pacity. The parameters M and K are different for each population.

Figure 2. Proportion of dispersing juveniles as a func-tion of distance. Data are from Atwood et al. (unpub-lished manuscript) for 1993 and 1994. The curve is the function y 5 0.4 · exp(2x/2.5).



Validation of the Habitat Function

Despite the highly significant fit of the data, and the ob-vious difference between the distribution of habitat suit-ability (HS) values at all locations and at locations withgnatcatcher observations, this analysis does not providean independent validation of the habitat function be-cause these are the observations we used in the statisti-cal analysis. One method of validation is to separate thecurrent observations in two and use only one set for esti-mation and the other set for testing. We separated theobservation locations and random points into two sets,dividing the map roughly in half with all coastal observa-tions in the southern half, with all inland observations inthe northern half. We re-estimated the habitat functionwith the northern half of the data set, which gave aslightly different equation from the one estimated with

the whole data set. We used this equation to predict thehabitat suitability values of observed pair locations in thesouthern half of the data set. Because these observationswere not used in the estimation of the function, theycan be used to validate the model.

The results of this validation (Fig. 5b) show the fre-quency distribution of the habitat suitability values of lo-cations with gnatcatcher pairs in the south, based on thehabitat suitability function estimated with the northernhalf of the data set. Except for the small sample size (be-cause only half the data are used), the results are similarto those in the previous figure: a large proportion (90%)of the observations are in cells with predicted habitatvalues of 0.5 or more. The habitat function estimatedfrom the northern half successfully predicted the obser-vations in the southern half.

We made two additional types of validation. We usedthe function estimated with the location of pairs to pre-

Figure 4. Frequency distribution of habitat suitability (HS) values for California Gnatcatchers in the study area. The HS values were calculated for each of 85,333, 1-ha cells in the study area, with a habitat function that was estimated by regression of presence-absence data on habitat variables.

Figure 5. Frequency distribution of the habitat suit-ability (HS) values of locations where California Gnat-catcher pairs were observed in all of the study area (a) and in the southern part of the study area (b). The HS values in (b) are predicted using data only from the northern half of the data set, and thus are used as one of three validations of the habitat function.

Table 3. Results of logistic regression for habitat function of California Gnatcatcher.*

Variable* Regr. coefficient SE Wald x2

p

CSS 6.7494 · 1024 1.6781 · 1024 16.1762 0.0001ELV 28.9122 · 1023 2.4068 · 1023 13.7118 0.0002DGR 22.6755 · 1023 0.5961 · 1023 20.1444 0.0001DTR 21.8863 · 1023 0.3759 · 1023 25.1813 0.0001CSS · ELV 218.3337 · 1027 5.8438 · 1027 9.8427 0.0017CSS · DTR 4.2860 · 1027 1.2153 · 1027 12.4385 0.0004ELV · DTR 8.5879 · 1026 2.6093 · 1026 10.8326 0.0010constant 1.8223 0.5797 9.8806 0.0017

Goodness of fit (chi-square for covariates):Log likelihood statistic 5 523.783 with 7 df (p 5 0.0001)Score statistic 5 374.544 with 7 df (p 5 0.0001)

*The habitat variables are coastal sage scrub (CSS), elevation (ELV),distance from grassland (DGR) and distance from trees (DLR). Slope,aspect, and distance from wetlands were non-significant.



dict the HS values of locations of single birds. About 89%of the single locations were predicted to have a HS valueof 0.5 or more. We used a function estimated with thelocation of pairs in 1992 to predict the HS values of loca-tions of pairs in 1994. About 82% of the pair locations in1994 were predicted to have a HS value of 0.5 or more.The habitat function estimated with observations ofpairs successfully predicted the observations of singles,and the habitat function based on 1992 pairs success-fully predicted the observations of 1994 pairs.

Patch Structure

Given the habitat map and the (medium) parameter esti-mates described above the program found 13 habitatpatches (clusters of suitable cells within the neighbor-hood distance of each other). The 2 largest patchesmade up about 84% of the total area of all patches (Table4, Fig. 6). Not all cells within the patch boundaries aresuitable at the threshold level of 0.5, as discussed above.In other words, the patches have “holes” in them, asrepresented by the lighter gray regions within patches(Fig. 6). Such areas are not counted in the area ofpatches, and the habitat values in these areas are notused in the calculation of carrying capacities (K) or ini-tial abundances (N0) reported in Table 4. The total carry-ing capacity was 3357 birds, or (at stable distribution)629 adult pairs. The total initial abundance was 2952birds, or 553 adult pairs.

Viability

With most parameter combinations, the model pre-dicted a fast decline and a high risk of extinction of thegnatcatcher populations. With medium estimates of allvariables, the risk of falling below the metapopulationthreshold of 60 individuals was about 19% in 20 years

and about 76% in 50 years. The median time to fall be-low the threshold was about 34 years. The sensitivitiesof these three results to each parameter are listed in Ta-ble 5. These sensitivity results are taken from cumulativetime to decline curves (as in Fig. 7). The results weremost sensitive to the probability of weather-related ca-tastrophes, and also very sensitive to adult fecundity, ju-venile fecundity, and parameters related to density de-pendence (the type of density dependence, and Alleeeffects that describe density effects at low abundances).All five parameters caused more than 10 years differencein median time to extinction and/or a difference of morethan 0.2 in risk of extinction (falling below the metapop-ulation threshold) in 50 years.

The results showed moderate sensitivity to seven otherparameters: juvenile survival, adult survival, metapopula-tion extinction threshold, threshold habitat suitability,carrying capacities, standard deviation of fluctuations in vi-tal rates, and density dependence in dispersal. The resultswere not sensitive to neighborhood distance, initial abun-dances, dispersal distance, correlation among populationfluctuations, and the number of fires.

Table 4. Carrying capacities (K), initial abundances, and areas of the patches identified by the model with medium valuesof parameters.

Rank KInit.

abund.Area

(km2)Area as percentage

of patches

1 1721 1514 101.0 49.822 1130 994 68.8 33.923 133 117 8.2 4.034 124 109 8.8 4.335 72 63 4.3 2.106 39 34 3.1 1.517 36 32 2.5 1.218 32 28 2.1 1.059 32 28 2.0 0.99

10 13 11 0.7 0.3511 11 10 0.6 0.2812 7 6 0.4 0.1913 7 6 0.4 0.21

Total 3357 2952 202.7 100.00

Figure 6. Patch structure of the model superimposed on the habitat suitability map of California Gnat-catcher in Orange County. Shades of gray represent habitat suitability (the darker the color, the higher the suitability), the black outline is the border of the study area, and the thin, white outlines are the outer bor-ders of patches.



Discussion

The results of three types of validation of the habitatsuitability function suggest the function is a good estima-tor of the quality of habitat in occupied locations. How-ever, the major weakness of this function is the lack ofdata on absences (locations where gnatcatchers are ab-sent), which forced us to use random locations. Thismight have caused an overestimation of habitat qualityin certain locations. If such data become available, thefunction may be narrowed, excluding some of the areasnow included as habitat in the patches. Another limita-tion of the habitat map is its geographic coverage. Thecoastal sage scrub in the study area may be connected tosimilar habitat in southern Orange county and else-where. One of potential improvements to the model in-volves expanding it to include the populations of Cali-fornia Gnatcatcher in other areas.

One of the specific aims of this analysis was to demon-strate how uncertainty and variability can be incorpo-rated in models for determining the threat a speciesfaces. We incorporated natural variation (resulting fromtemporal fluctuations in environmental factors) in theform of randomly distributed vital rates (survivals and fe-cundities) and as natural catastrophes. We also modeleddemographic stochasticity in reproduction, survival, anddispersal. We used these types of natural variation (envi-

ronmental and demographic) to express the model re-sults in probabilistic terms (e.g., risk of decline).

Most parameters are not precisely known because ofmeasurement errors and lack of data. In such cases, wemade three estimates of the parameter; estimating lowerand upper bounds, in addition to a “medium” estimate.We thus incorporated uncertainty that results from lackof knowledge in the form of parameter ranges. We usedthese ranges to estimate upper and lower bounds on theestimated viability of the species. In other words, weused natural variability to estimate risks and uncertain-ties to estimate the uncertainty of these risk estimates.

The sensitivity of results to catastrophe probability isnot surprising. Because we did not have enough data tocorrelate vital rates with weather-related variables, weused a wide range for this parameter. The deterministic,long-term growth rate (eigenvalue, l) predicted by thestage matrix (with medium values of vital rates) wasabove 1.0, and declines and extinctions occurred mostlydue to catastrophic declines in the vital rates, as well asyear-to-year environmental fluctuations and Allee effects.The declines predicted by the model also explains theinsensitivity of metapopulation viability to fires. We mod-eled fires as causing a decrease in carrying capacity (K),but the populations were most of time well below K andthus were not affected by a decline in K caused by fires.

A surprising result was the negative (but weak) effect

Table 5. Sensitivity of model results to parameters: difference in median time to decline and probability of decline between models with high and low values of each parameter.

Parameter Effect a

Median time todecline (yr)

Prob. of declinein 20 years

Prob. of declinein 50 years Max. diff.

(year)dabsb percc absb percc absb percc

Threshold HS 2 8.7 25.7% 0.0837 44.2% 0.1617 21.4% 41Neighborhood distance 1 3.1 9.2% 0.0382 20.2% 0.0604 8.0% 35Carrying capacity 1 6.7 19.8% 0.0857 45.2% 0.1119 14.8% 35Initial abundance 1 3.0 8.9% 0.0656 34.6% 0.0307 4.1% 26Dispersal distance 2 5.2 15.4% 0.0502 26.5% 0.1119 14.8% 41

Juvenile survival 1 7.5 22.2% 0.0771 40.7% 0.1452 19.2% 42Juvenile fecundity 1 11.8 34.9% 0.1056 55.7% 0.2332 30.8% 38Adult survival 1 8.4 24.9% 0.0875 46.2% 0.1662 21.9% 38Adult fecundity 1 14.7 43.5% 0.1452 76.6% 0.2888 38.1% 40

Density dependence 1 14.7 43.5% 0.0812 42.8% 0.2393 31.6% 47Allee effects (local threshold) 2 10.3 30.5% 0.1605 84.7% 0.1627 21.5% 34Density dependent dispersal 1 7.2 21.3% 0.0719 37.9% 0.1278 16.9% 36

Standard deviations 2 7.8 23.1% 0.0914 48.2% 0.1205 15.9% 33Correlation 2 2.8 8.3% 0.0402 21.2% 0.0334 4.4% 33Catastrophe (weather) prob. 2 61.3 181.4% 0.6546 345.4% 0.7342 96.9% 32Number of fires 2 0.5 1.5% 0.0119 6.3% 0.0160 2.1% 43Metapopulation threshold 2 7.2 21.3% 0.1204 63.5% 0.1021 13.5% 31aEffect on metapopulation viability. Indicates whether high parameter value resulted in higher (1) or lower (2) viability. For example, higheradult fecundity resulted in higher viability, and the difference in median time to fall below the threshold between high and low estimates ofadult fecundity was 14.7 years. This corresponded to 43.5% of the median time (33.8 years) with the medium estimate of all parameters, includ-ing adult fecundity. The difference in risk (of falling below the metapopulation threshold of 60 individuals within 20 years) between the lowand high adult fecundity was about 0.145, or 77% of the risk (0.189) with the medium estimate.bAbsolute difference between the results of the two models with high and low values of the parameter.cDifference as a percentage of the result with the medium value of the parameter.dYear at which the difference in risk of decline (with the high and low values of the parameter) was the largest.



of dispersal distance on viability, which can be ex-plained by a type of source-sink dynamics. Although allpopulations in our model had the same vital rates (hencethe same l), the smaller populations were more proneto extinction as a result of demographic stochasticity. In-creased dispersal distance meant a larger number of dis-persers going from the more stable larger populations tothese smaller and more extinction-prone populations.To demonstrate that the smaller populations indeed actas sink populations, we ran two other simulations, withthe ninth largest and fifth largest populations, by simplydeleting the smaller populations. The results of thesesimulations showed that when the smaller populationswere excluded, the risk of decline to any level waslower (Fig. 8). We made this comparison only to demon-strate that small populations can act as sink populations,even if their vital rates are the same as the larger popula-tions. However, this result should not be interpreted interms of reserve design because we also assumed therewas no dispersal to the patches we deleted from themodel (i.e., dispersal was only to the patches that re-mained in the model). There is, of course, no guaranteethat gnatcatchers would not attempt to disperse to theseareas whether or not they are part of a reserve system.

Increased dispersal had a similar effect of increasingextinction risk in another model: Akçakaya and Baur(1996) concluded that when some populations weremodeled as being subject to more severe catastrophesthan others, it created a type of source-sink dynamics,

which caused this effect. When catastrophes wereevenly distributed, increased dispersal caused a decreasein extinction risks.

Two of the model parameters, threshold habitat suit-ability and metapopulation extinction threshold, havenecessarily arbitrary values. The rationale for selectingany value for these thresholds is similar to the one usedin deciding a probability level for statistical significance.Just as the level of statistical significance depends on pri-orities (e.g., whether one wants to minimize Type I orType II error), so do these parameters. Threshold habitatsuitability reflects a compromise between making surethat no suitable habitat is left out and making sure thatunsuitable habitat is not included in patches. Similarly,the extinction threshold depends on whether it is moreimportant not to overestimate or not to underestimateextinction risks. It is therefore positive that the resultswere only moderately sensitive to these parameters.

The results point to a need to better estimate twogroups of ecological parameters. One group is the vitalrates (especially fecundity) and the frequency andamount of change in vital rates caused by catastrophes.The other group includes density dependence parame-ters, including Allee effects. Detailed data on vital ratesmay also help estimate these parameters, for example bycomparing fecundities in regions with different densityof gnatcatchers. Such a data set may also help link the vi-tal rates to local habitat suitability, eliminating one of thestronger assumptions of our model (that vital rates arethe same for all populations).

The sensitivity of results suggests that the resultsshould not be interpreted in absolute terms. Specifically,it would be inappropriate to use the results of this

Figure 7. Sensitivity of risk of decline to adult fecun-dity (Fa ). The curves show the probability of falling be-low the metapopulation threshold as a function of time. The vertical dashed lines show the median time to decline. The arrows show four sensitivity results: the difference between median time to decline (A), the dif-ference in risk of decline in 20 years (B) and 50 years (C), and the maximum difference in risk of decline (D), which in this case occurred around year 40.

Figure 8. Risk of decline (by 80% to 100% from the initial metapopulation abundance) predicted by mod-els with 13, 9, and 5 populations. The vertical line marks the metapopulation extinction threshold of 60 individuals.



model to conclude that gnatcatcher populations in Cen-tral/Coastal Orange County are either threatened by ex-tinction or secure from such a threat. There is simplytoo much uncertainty about most of the parameters topredict with confidence what the population size willbe in 50 years or what the risk of extinction might be.Despite this uncertainty, we believe the model can havepractical use in two areas.

The sensitivity analysis described is one potential useof modeling in the conservation and management of thecoastal sage scrub community; it gives informationabout which parameters need to be estimated morecarefully. Once a model is developed, it can be im-proved with such analyses carried out at least once ayear to incorporate new data collected during that year.

The second use of such a model is to rank manage-ment options in terms of their predicted effect on the vi-ability of target species. Results of population viabilitymodels such as this one are less reliable if interpreted asabsolute predictions than if interpreted as relative toother sets of assumptions or scenarios. Even if results ofa model are sensitive to various model parameters, it ispossible that relative rankings of management optionswill not be as sensitive. In other words, the specific pre-dictions of the model may change as a function of pa-rameters and assumptions, but all assumptions may stillpredict that a particular management option ranks bet-ter than its alternatives in terms of the viability of thespecies. This would make such a relative ranking a muchmore reliable prediction than a prediction of exactlywhat the population size would be 50 years from now.

In principle, all possible management actions can berepresented as changes in habitat suitability or demo-graphic parameters (including their variances and the ini-tial values of variables such as abundances in each stageand each population). The consequences of these param-eter sets can be estimated by the model in terms of the vi-ability of the species and then used to rank alternativemanagement actions, to prioritize conservation measures,and to evaluate the importance of different parameters.

The uncertainty of model results depends on the timehorizon of the simulation. The cumulative time to quasi-extinction results showed that the greatest differencebetween the results given by different parameter combi-nations occurred around 30 to 40 years. This suggeststhat the results are most sensitive to changes in parame-ters when the time horizon is only a few decades. Thismight be an appropriate time horizon if the model isused in the future to compare alternative managementoptions. Longer or shorter time horizons may underesti-mate the effects of management actions because themodel results may not be as sensitive to changes in pa-rameters (as a function of the simulated effect of themanagement action). As more demographic and ecologi-cal data accumulate, it will become possible to make as-sessments with longer time horizons.

Acknowledgments

This study was funded by Southern California Edison.We thank J. Palmer for making this collaboration pos-sible, R. Downer for his help in the initial phase of theresearch, and M. Lebben for his help in organizing the GISdata. J. L. A.’s involvement was financially supported bythe trustees of Manomet Observatory for ConservationSciences, Southern California Edison, and the SuperparkProject. D. Bontrager and E. Almanza provided valuablediscussion of gnatcatcher biology in Orange County. S.Pimm and two anonymous reviewers made helpful com-ments and suggestions.

Note Added in Proof

Various data in this paper are from submitted, but as yetunpublished, manuscripts. These papers are availablefrom the authors.

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