SPATIAL OPTIMIZATION OF PRAIRIE DOG COLONIES FORBLACK-FOOTED FERRET RECOVERY

MICHAEL BEVERS and JOHN HOFRocky Mountain Forest and Range Experiment Station, Fort Collins, Colorado

DANIEL W. URESKRocky Mountain Forest and Range Experiment Station, Rapid City, South Dakota

GREGORY L. SCHENBECKNebraska National Forest, Chadron, Nebraska

(Received October 1995; revision received July and August 1996; accepted January 1997)

A discrete-time reaction-diffusion model for black-footed ferret release, population growth, and dispersal is combined with ferretcarrying capacity constraints based on prairie dog population management decisions to form a spatial optimization model. Spatialarrangement of active prairie dog colonies within a ferret reintroduction area is optimized over time for maximum expected adultferret population. This modeling approach is applied in an exploratory case study to a black-footed ferret reintroduction program inBadlands National Park and Buffalo Gap National Grassland, South Dakota. The model is currently being used to evaluate prairiedog population management alternatives and captive-bred ferret release locations for the Buffalo Gap National Grassland. Thisapproach is also being adapted for use on other grasslands and with other species in the northern Great Plains.

Early in 1987 the black-footed ferret (Mustela nigripes)became one of the world’s most endangered mam-

mals when the last known free-ranging member of thespecies was taken into captivity (Thorne and Belitsky1989). The Wyoming Game and Fish Department was suc-cessful in breeding six of the surviving ferrets in captivity(Clark 1989). This set the stage for a national recoveryprogram of releasing captive-bred ferrets back into thewild.

Historically, the black-footed ferret ranged sympatricallywith prairie dogs (Cynomys sp.) across much of NorthAmerica (Anderson et al. 1986). Available evidencestrongly supports the conclusion by Henderson et al.(1969) that black-footed ferrets have narrow habitat re-quirements, living principally in prairie dog burrows anddepending primarily on prairie dogs for prey (Linder et al.1972). Demise of the species in the wild has been attrib-uted to loss and fragmentation of habitat (prairie dog col-onies) due to extensive prairie dog eradication programsand changes in land use, combined with susceptibility ofprairie dogs to sylvatic plague and of ferrets to caninedistemper (U. S. Fish and Wildlife Service et al. 1994). AsSeal (1989) points out, it now appears difficult to find suit-able ferret habitat complexes (“groups of prairie dog colo-nies in close proximity,” Biggins et al. 1993) of 3,000 to

15,000 hectares, even though prairie dogs were once dis-tributed over 40 million hectares of land.

The first release of captive-bred black-footed ferrets intothe wild occurred in 1991 in Shirley Basin, Wyoming. Twoadditional reintroduction areas were added in 1994, in-cluding the site of this study centered in Badlands NationalPark, South Dakota. These ferret release sites were selectedon the basis of habitat suitability and other biological andsociopolitical factors. Prairie dog population managementwithin these sites will be a critical component in the suc-cess or failure of ferret reintroductions. Rodenticides areactively employed in the northern Great Plains, and havegreatly reduced prairie dog populations (Roemer and For-rest 1996). Black-footed ferret recovery at the Badlandsreintroduction site will likely be affected by the locationand timing of rodenticide treatments in the area.

The spatial arrangement of prairie dog colonies in acolony complex has important effects on the number ofblack-footed ferrets that can be supported (Minta andClark 1989). As prairie dog colonies become smaller ormore widely separated, successful ferret dispersal betweencolonies is less likely and the total population that can besupported is reduced. Houston et al. (1986) and Miller etal. (1988) have employed spatial measures such as meanintercolony distance and colony size frequency distribution

Subject classifications: Natural resources: habitatming, linear, applications:

allocation for endangered species recovery. Probability,a black-footed ferret case study from South Dakota.

diffusion: dynamic spatial population optimization. Program-

Area of review: OR PRACTICE.

Operations Research 0030-364X/97/4504-0495 $05 .OOVol. 45, No. 4, July-August 1997 495 0 1997 INFORMS

496 / BEVERS, HO F, URESK AND SCHENBECK I

in estimating ferret habitat suitability, but Biggins et al.(1993) note a number of troubling quantitative difficultieswith such approaches. For example, it is often possible toidentify a number of habitat patch arrangements that areequal in mean intercolony distance (as well as individualand total patch sizes) for which expected population re-sponses would typically not be equal according to biodiffu-sion theory (Okubo 1980), island biogeography theory(MacArthur and Wilson 1967), or metapopulation models(e.g., Hanski 1994). Instead, estimates of pairwise dispersalbetween habitat patches are generally considered impor-tant. Consequently, in the Biggins et al. procedure, theeffects of spatial colony arrangement within colony com-plexes are assessed qualitatively.

This paper presents the development of a more rigorousquantitative approach that is adapted from the theoreticalspatial optimization work of Hof et al. (1994). In thatapproach, the ability of a modeled wildlife population togrow and expand was constrained by a combination of theinherent reproductive and dispersal characteristics of the spe-cies, and the size and arrangement of habitat, imposed aslimiting factors. Hof et al. used binary habitat managementvariables to determine the resulting arrangement and car-rying capacity conditions on a homogenous landscape overtime. A continuous variable approach based on these ideasis presented here for black-footed ferret recovery. Themodel is used to explore habitat management and ferretrelease as a spatial efficiency problem on the federallymanaged lands of the Buffalo Gap National Grassland ad-jacent to the Badlands National Park ferret release area.

1. THE SPATIAL OPTIMIZATION MODEL

Within the reintroduction area, black-footed ferret habitatcomprises a complex of active and potential prairie dogcolonies (patches) forming distinct habitat “islands” on thelandscape. The approach described here is related to ear-lier biodiffusion models (Skellam 1951, Kierstead and Slo-bodkin 1953), and island biogeography models (Allen1987). Using a continuous reaction-diffusion equation for asingle habitat patch with exponential population growth,Kierstead and Slobodkin established a critical patch sizebelow which the population perishes. By discretizing thehabitat into a number of individual patches, or islands-each too small to individually support a persistent popula-tion-Allen proved several important theorems, includingthe existence of a critical number of patches in a lineararrangement of such islands, below which the populationagain perishes. Our spatial optimization model retainsthese characteristics but uses discrete time periods andapproximates habitat patch configurations with a grid ofcells on the landscape. We then incorporate cellular habi-tat management decision variables so that all potentialspatial configurations can be considered (within the reso-lution of our grid). With these decision variables to controlthe amount and location of ferret carrying capacity overtime, our reaction-diffusion model can go beyond simply

evaluating persistence or extinction to estimating expectedpopulation size. Annual time periods also more closelymodel key ferret life history processes.

Ferret population growth and dispersal between cellsfrom year to year is modeled here with an exponentialpopulation growth potential and a random dispersal pat-tern that relates probability of dispersal to distance. Withdiscrete spatial cells and time periods, this reaction-diffusion process can be captured with linear constraints,as described below.

Rodenticide treatments, which have a negative effect onblack-footed ferrets by reducing prairie dog numbers, arethe principal habitat management action to be considered,as rodenticide applications are expected to continue on theNational Grassland. Thus, a tradeoff exists between ex-pected ferret population and the level (location, timing,and amount) of rodenticide use employed. Particular fer-ret habitat layouts are achieved over time by the prairiedog populations that result from the rodenticidetreatment-nontreatment schedules applied to each cellacross the landscape, on the premise that prairie dog pop-ulations will recover rapidly in areas left untreated. Forany proposed habitat layout, the model is a useful methodfor estimating the expected ferret population over time.Beyond that, the model is also useful for finding efficienthabitat layouts under various habitat (prairie dog popula-tion) policy constraints, so that the maximum number offerrets can be supported given those constraints.

Decision variables (Xihk) are defined for each possibleschedule (indexed by k) of annual rodenticide treatment ornontreatment in each habitat condition class (indexed byh) for each cell (indexed by i). For example, one schedulecould call for rodenticide treatments in the first year andevery fourth year thereafter, while another schedule mightcall for treatments to begin in the second year instead. Athird schedule could impose no treatments at all. Based onthe number of hectares assigned to each habitat manage-ment decision variable (schedule Xihk), adult black-footedferret carrying capacity for a cell in any given year is thenestimated from the prairie dog population expected underthat management schedule. We will assume that no prairiedog populations would occur outside the selected habitatareas, although in practice, small numbers of prairie dogscan generally be expected.

Adult black-footed ferret populations expected in eachcell in any year are limited by either the carrying capacityof that cell, or by the ability of ferrets from nearby cells tosuccessfully reproduce and disperse there, or both. Addi-tional decision variables (Rit) are used to determine thetiming (year t) and location (cell i) for captive-bred ferretreleases into the area. Release locations selected by themodel are useful because locations for available ferrethabitat are simultaneously scheduled. Due to their rapiddispersal, ferrets must be released in areas where ampleprairie dog populations occur in the surrounding area aswell as within the immediate vicinity in order to survive.The solution of the model indicates a complex of prairie

the next 10-15 years. Under current management plans,Buffalo Gap National Grassland supports an estimated2112 ha of predominantly active prairie dog colonies re-served from rodenticide use adjacent to Badlands NationalPark. We refer to these as “current” colonies. The Grass-land contains an additional estimated 9850 ha of predom-inantly inactive prairie dog colonies in the study areawhich have been treated with rodenticide in past years. Werefer to these as “potential” colonies.

Spatial Definition

We selected U. S. Public Land Survey sections as cells forthe model and assumed for dispersal probability calcula-tions that each of the 608 survey sections (indexed by i) inthe study area was a square enclosing 259 ha of land. Thenumber of hectares of existing prairie dog colonies withineach section were estimated from color infrared aerialphotography taken in August of 1993 using methods de-scribed by Schenbeck and Myhre (1986) and Uresk andSchenbeck (1987). Active prairie dog colony areas withinthese intact burrow systems were inventoried from fieldsurvey records. Inactive areas were identified as readilyrecoverable for the next 10-15 years, along with areashaving intact burrow systems identifiable in similar aerialphotographs taken in 1983. The 1983 prairie dog colonydistribution was used to estimate potential colony distribu-tion because this was the period when recorded prairie dogpopulations were greatest. Other suitable prairie dog hab-itat areas lacking burrow systems since 1983 were not in-ventoried for this model, under the assumption thatpopulation establishment in those areas is beyond the10-15 year time frame of interest. Land areas within eachsurvey section were classified as either National Park Ser-vice administered lands (h = 1), USDA Forest Serviceadministered lands presently subject to prairie dog popula-tion control (potential, h = 2), or USDA Forest Serviceadministered lands presently reserved from prairie dogpopulation control (current, h = 3). Privately owned landswere not included in the model.

Ferret Dispersal

Although relatively few observations of ferret movementsare available, distances of 2-3 km were typical for bothnightly movements and annual intercolony movements(primarily by juveniles in late summer or early autumn) ofwild-born ferrets at Meeteetse, Wyoming (Forrest et al.1985, Biggins et al. 1986, Richardson et al. 1987). Thelongest nightly move reported from that complex is about7 km. Oakleaf et al. (1992, 1993) report substantiallylonger dispersal distances (up to 17.5 km) over the first 30days following captive-bred ferret releases at the ShirleyBasin prairie dog colony complex in Wyoming. The statis-tics reported by Oakleaf et al. roughly suggest an exponen-tial distribution of dispersal distances, while dispersal wasapparently equally likely in all directions (although fewobservations are available). Eight of the ferrets released inBadlands National Park in 1994 were observed to disperse

BEVERS, H O F, URESK AND SCHENBECK / 499

with a mean distance of 3.7 km and a maximum distance of11.8 km (standard deviation = 4.2 km) over about a 30-dayperiod. It is not known to what degree differences betweenthese observations result from differences between captive-bred and wild-born ferrets, differences between prairie dogcolony complexes, or from other causes.

For this study, we assumed that all ferrets will disperseannually according to an exponential distance distributionwith a mean of 3.7 km in uniformly random directions overa radius of about 14 km. Dispersal coefficients (gji) werethen estimated by numerical approximation of the integralof this bivariate dispersal distribution over distances andangles defined by the boundaries of each destination (i)section relative to the center of each source (j) section.The effects of rugged topography in the Badlands could betaken into account in the pairwise estimation of dispersalcoefficients, but these effects are unknown.

Net Population Growth Rate

Wild ferrets have not been studied under conditions ofunlimited habitat. Consequently, values for rj were esti-mated by simulating unlimited population growth usingmean birth and death rates and initial conditions similar tothose assumed by Harris et al. (1989) in their research onblack-footed ferret extinction probabilities. Beginningfrom expected values of 1 male (yearling or older), 1 year-ling female, and 1.2 adult females (two years old or older),the simulated population was iteratively grown year byyear. Yearling females were expected to produce 0.85 lit-ters each, while adult females were expected to produce0.95 litters each. Each litter was expected to produce 1.7juvenile males and 1.7 juvenile females. Mortality thenremoved half of the juvenile males, 40 percent of the juve-nile females, 20 percent of the adult males, and 10 percentof the adult and yearling females. After 12 simulationyears the population ratios and growth rates stabilized withan r-value (annual net population growth rate) of 0.8175.

Ferret Releases

Based on past experiences (Oakleaf et al. 1992, 1993),approximately 80 percent of released captive-bred black-footed ferrets are expected to die during their first 30 daysin the wild. Half of the remaining ferrets are expected toperish during their first winter. Of the 36 ferrets releasedin Badlands National Park in 1994, only eight were knownto survive the first 30 days. Taking into account likely win-ter mortality, we assigned an expected population value of0.5 adult ferrets at each of the eight surviving ferret loca-tions as initial conditions (Ni) in the model (with zeroesassigned elsewhere).

We expected 40 more ferrets to be released in the fall ofeach of the following four years. Assuming that four fer-rets from each release would survive to reproduce, we setb1 - b4 equal to 4.0 (with zeroes assigned for all otheryears).

500 / BEVERS, HOF, URESK AND SCHENBECK

BadlandsNational Park

Buffalo Gap National Grassland

Map Code q= (0, I] Black-Footed Ferret Capacityq= (1, 4]

= (4,7]n= (7, 10]n= (10, 13.66)

Figure 1. Available short-term (10-15 year) adult black-footed ferret carrying capacities on federally managed lands withinthe study area.

Ferret Carrying Capacity

Although prairie dog densities within colonies typically de-cline over extended periods of time (Cincotta 1985,Hoogland et al. 1988), existing colonies in the Badlandsarea for the next 10-15 years are expected to remain wellabove the lower limit of “good” ferret habitat (3.63 prairiedogs/ha) estimated by Biggins et al. (1993). Consequently,we based our estimate of maximum ferret carrying capacityon the observations reported by Hillman et al., (1979) offerret populations in Mellette County, South Dakota. Inthe model, adult ferret carrying capacities (Cihkt) on exist-ing or fully recovered prairie dog colonies were set at0.05273 ferrets/ha. Figure 1 shows the spatial arrangementof current plus potential ferret habitat by survey section inthe study area at maximum model carrying capacity (deter-mined by summing 0.05273 Aih across h for each section i).Carrying capacity for the entire area was about 757 adultferrets. Most of the ferret carrying capacity shown on theNational Grassland (outside of the Park boundary) is po-tential rather than current habitat, comprising predomi-nantly inactive prairie dog colony burrow systems. Thisdoes not necessarily inhibit ferret establishment on theGrassland in our model, however, because prairie dog popu-lations can generally recover more quickly than ferretpopulations can be established.

Potential prairie dog colonies are not presently able tosupport ferrets at maximum carrying capacity due to pasttreatments with rodenticide. Based on studies by Knowles(1985), Cincotta et al. (1987), and Apa et al. (1990), weestimated that complete prairie dog population recovery inrecently treated colonies would require an average of four

breeding seasons. We set adult ferret carrying capacity(cihkt) accordingly at one-eighth of full capacity (0.00659ferrets/ha) for the first year following use of rodenticide, atone-fourth of full capacity (0.01318 ferrets/ha) for the sec-ond year, at one-half of full capacity (0.02636 ferrets/ha)for the third year, and at full capacity thereafter (given noadditional rodenticide treatments). This rate of recoverycould require special management actions, such as inten-sive livestock grazing, to aid the spread of prairie dogs(Uresk et al. 1981, Cincotta et al. 1988). We also assumedthat all potential habitat areas in the model could beginrecovery in any chosen year.

3. MODEL RESULTS

The model was solved on a personal computer using Ket-ron's commercial linear programming package “C-Whiz,”with Equation (9) as the objective function. Equation (8)was used in six separate optimizations with different right-hand side (Cpt) levels to restrict the amount of ferret car-rying capacity added from potential prairie dog colonieson the Grassland (hh = 2) to form a tradeoff analysis. Thus,for each of the six alternatives, a single policy constraint

(p = 1) was used with identical right-hand-side amountsfor each year t. The Xi2k decision variables were givennonzero Cihktp coefficients for years in which no rodenticidetreatments were scheduled. All other decision variableswere given Cihktp coefficients of zero. A 25-year planninghorizon (T = 25) was used to allow enough time for ex-pected ferret population levels to stabilize, but care mustbe taken not to overinterpret results beyond 10-15 years.

BEVERS, HO F, URESK AND SCHENBECK / 501

600

00 1 2 3 4 5 6 7 8 910111213141516171819202122232425

Year

- - -. Current Park and . . . . . . . . . . . Plus 20% of Potential - - Plus 40% of PotentialGrassland Habitat Grassland CarryingAreas Capacity

Grassland CarryingCapacity

. . l Plus 60% of Potential -.-.- Plus 80% of Potential _ Plus 100% of PotentialGrassland Carrying Grassland CarryingCapacity Capacity

Grassland CarryingCapacity

Figure 2. Expected adult black-footed ferret populations under the present management strategy and five alternativestrategies.

Beyond 15 years, additional potential prairie dog habitatmight need to be considered, as well as the possibility ofdeclines in prairie dog population densities for oldercolonies.

Figure 2 shows the total expected adult ferret popula-tion (Ft) resulting from allowing no additional carryingcapacity (Cptt = 0), and from five 20-percent increments ofhabitat (capacity for 103.88 additional adult ferrets perincrement) from the potential Grassland prairie dog colo-nies. The lowest expected population curve in Figure 2results from using rodenticides to prevent any increase inprairie dog populations from current levels (Cpt = 0). Thehighest expected population curve results from discontinu-ing rodenticide use in the area altogether (Cpt = 519.4).Due to ferret dispersal, increments of additional ferretcarrying capacity do not result in proportional increases inexpected ferret population.

In all cases in Figure 2, sigmoidal expected populationgrowth curves resulted. As we would expect, the graphshows diminishing marginal returns as more carrying ca-pacity is added because the most spatially efficient habitatareas are included first. Also, each curve levels off substan-tially below total allocated carrying capacity. For example,when all Grassland habitat areas are allocated to prairiedog colonies, the expected population of ferrets rises to

only about 85 percent of the summed capacity of morethan 757 adult ferrets. This suggests that simply totallingavailable carrying capacity will tend to overestimate thepopulation size that can be supported because spatial ar-rangements are not taken into account.

Preferred habitat areas change over time, as shown inFigure 3 by maps of the habitat allocated in different years(expressed as adult ferret capacity, calculated by summingcihktXihk across h and k for each i and t) under the alter-native which adds 20 percent of the potential Grasslandcarrying capacity for ferrets. The 20-percent limit for thisalternative is binding from year 7 on. Prior to that year,the expected ferret population is still small enough that theconstraint is not limiting. Figure 3a shows the habitat allo-cations for year 7. Survey sections outside of the Park andcurrent Grassland colony areas (outlined in the figure) arepotential habitat, where management choices (schedules ofrodenticide treatment) were allowed. In Figure 3a, a smallamount of habitat is allocated to all but one survey sectionwith potential habitat because the fledgling expected ferretpopulation is rapidly expanding throughout the area (com-pare with Figure 1). The expected population in year 7 (Si7for each section i) is shown in Figure 4b, along with thecorresponding selected ferret releases (the sum of Ritacross t for each section i) shown in Figure 4a.

504 / BE V E R S, HO F, U RESK AND S C H E N B E C K

By year 15, the expected ferret population under thisalternative has largely leveled off at more than 230 adultferrets, and the preferred habitat allocations have shiftedto more concentrated areas around the Park and currentGrassland colonies (Figure 3b). Many survey sections‘which had some habitat allocated in year 7 no longer haveany habitat allocated by year 15. The corresponding ex-pected ferret population (Si15 for each section i) is shownin Figure 4c.

Based on these results, we examined the habitat alloca-tions for this alternative near equilibrium conditions bybounding all ferret release variables to zero (Rit in Equa-tion (5)), unbounding all initial population variables (Si0 inEquation (3)),, and using Equation (1) as the objectivefunction. With initial populations no longer restricting thesolution, a static equilibrium can be approximated withoutthe need for building a different model. We also allowed themodel to schedule treatments for Grassland colonies cur-rently reserved from rodenticide use (h = 3) in addition topotential colony (h = 2) treatments for meeting the policyconstraint to allow greater spatial freedom for habitat se-lection. The resulting allocations (Figure 3 c support anexpected population that levels off at a little more than 300adult ferrets. This is an increase of about 60 adult ferrets(in year 25) over the alternative that optimally combined20 percent of the potential Grassland with the currentprairie dog colonies (see Figure 2). The increase isachieved by exchanging some (but not all) of the prairiedog colonies from the current colony areas for new colo-nies in the nearby Grassland areas (compare Figures 3band 3c). While this near-equilibrium analysis appears toextrapolate beyond the supporting data due to the lengthof time involved, this may not be the case. If black-footedferret dispersal is actually a biased diffusion process, thepopulation might be able to take advantage of highly con-centrated habitat more quickly than our results wouldindicate.

The pattern of Grassland allocations in Figure 3c resultsfrom the interaction of two important effects. Many of thesections with the highest potential ferret capacity are leftunallocated by the model in order to round out the long,narrow habitat arrangements in portions of the Park. Inthe model, population losses from fully occupied habitatoccur from dispersal across the habitat perimeter into un-suitable areas, and from dispersal into areas already atcarrying capacity. For a given amount of habitat, carryingcapacity remains constant while dispersal losses into non-habitat areas can be lowered by changing the shape of thearea to reduce dispersal across the perimeter. Allocatingcircular patterns, which have the smallest perimeter-to-area ratio, would minimize losses and maximize retainedpopulation given uniformly random dispersal direction.This tendency appears to be compromised somewhat infavor of placing habitat close to as many sections of thePark as possible.

The results described thus far were obtained usingenough different management variables (coupled with

Table IThe Expected Number of Black-footed Ferrets inEach Year for the +20% Alternative under Three

Different Scheduling Formulations

Allocation One-Time Allocation Full SchedulingYear in Year 1 Change in Any Year Model

1 9 9 92 17 17 183 31 31 324 51 51 535 76 76 796 106 107 1147 139 139 1618 163 169 1779 182 183 190

10 196 195 20211 205 204 21212 212 210 21913 217 216 22514 220 219 22915 222 222 23216 224 224 23417 225 225 23618 225 226 23719 226 227 23820 226 227 23821 227 228 23922 227 228 23923 227 228 23924 227 228 24025 227 228 240

constraints on carrying capacity rather than on land alloca-tions) to provide a great deal of flexibility in ferret capacityplacement and timing through the selection of spatiallydefined rodenticide treatment schedules. However, suchflexibility may be impractical. Public land use planning fora 10-15 year period is often viewed as a process for sched-uling a one-time change (if any) in management. To testthe effects of that approach, we constructed a model inwhich the potential Grassland management variables(Xi2k) were redefined to form a more restrictive set ofoptions. Although the model could choose when to stoprodenticide treatments in a particular area (if at all), nofurther treatments could be scheduled afterward. In someplanning cases, scheduling is not even considered, and allmanagement changes take immediate effect. To test thoseeffects, we constructed another model in which the man-agement variables for each potential section (again, Xi2k)simply represented either repeated rodenticide treatmentsor none at all. Table I shows the yearly expected adultferret populations ( F t ) using these different approaches forthe alternative that allocates 20 percent of the potentialGrassland carrying capacity (in addition to the Park andcurrent Grassland colonies). Considering the small differ-ences in Table I, the use of simpler models (with greaterease of presentation and implementation) may not impactresults significantly.

The reductions in matrix size and complexity with thesimpler models were substantial. The full scheduling for-mulation included 20,946 rows and 65,598 columns, whilethe simpler “one-time change in management” schedulingmodel reduced the problem size to 17,795 rows and 21,972columns. The nonscheduling model further reduced thenumber of columns to 18,156.

4. CONCLUSION

Our model is the first application of this type of dynamicspatial optimization to a real-world problem of habitatevaluation and management design. With very limitedknowledge of ferret reproduction and dispersal in the wild,the model results must be regarded as an initial estimateof a lower bound on expected population levels for a givenhabitat arrangement. The method appears to be promisingin aiding with efficient design of alternative habitat man-agement and reintroduction strategies. The explicit ac-counting of spatial patch relationships, as opposed torelying on measures like mean intercolony distance, is thestrong point of the model. Viewing the model’s expectedpopulation estimates as lower bounds provides a usefulcontrast to results from habitat complex circumscriptionmethods (e.g., Biggins et al. 1993), which could probablybe viewed as estimates of expected population upperbounds, at least prior to any qualitative adjustments. Forspecies with highly random dispersal behavior, the spatialoptimization estimate of expected population should beespecially useful.

An evaluation of the Buffalo Gap National Grasslandprairie dog management and black-footed ferret recoveryprogram began in 1996 as part of a land managementplanning process for national grasslands across the north-ern Great Plains. This spatial optimization model andother published ferret models will be the principal toolsfor scientific review in that evaluation. Our model has al-ready been used to demonstrate the importance of moni-toring ferret dispersal, as well as survival and reproduction.Biologists are currently gathering data for refining and val-idating our model for future use, and are adapting themodel to other reintroduction areas. These reaction-diffusion methods have also been adopted for analyzinghabitat alternatives for other threatened and endangeredspecies in the northern Great Plains.

We must stress that the spatial optimization model isdeterministic, and produces estimates of expected popula-tion. Stochastic variation, which can be a particularly im-portant consideration at low population levels, is not takeninto account. Stochastic variation in net population growthhas been accounted for in a model developed by Harris etal. (1989) for estimating black-footed ferret population vi-abilities as functions of initial population size. Replacinginitial population size in their viability model with ex-pected population size from our spatial optimizationmodel would provide a lower bound estimate of expectedviability which takes into account stochastic variation in

BEVERS, HO F, URESK AND SCHENBECK / 505

net population growth. For some purposes, this might pro-vide an additional benefit by reducing results to aprobability-based index. However, stochastic variation ofother important model parameters remains unaddressed.

Clearly, there is much we do not yet know about black-footed ferret populations in the wild. More information onferret dispersal could be particularly useful. Consequently,the actual response of the new South Dakota populationwill likely be different from model predictions, at least interms of ferret densities. As more is learned through ferretmonitoring over the next several years, we anticipate thatthe model could be used as part of an adaptive manage-ment process (Walters 1986). The spatial optimizationmodel offers a great deal of flexibility for combining site-specific habitat and population information (as it becomesavailable) with management options and constraints. Fur-ther research is needed to understand the complexities offerret and other wildlife population dynamics, and to ac-count for those complexities in optimization modeling.

ACKNOWLEDGMENTS

The authors wish to thank Kieth Severson, Helen Fitting,and Peter McDonald for providing information and data,and Jill Heiner for computer programming used in thisstudy. The manuscript was substantially improved throughthe efforts of the editors and two anonymous referees.

REFERENCES

ALLEN, L. J. S. 1983. Persistence and Extinction in Single-species Reaction-diffusion Models. Bulletin of Mathemat-ical Biology, 45, 209-227.

AL L E N, L. J. S. 1987. Persistence, Extinction, and CriticalPatch Number for Island Populations. J. Math. Biology,24, 617-625.

ANDERSON, E., S. C. FORREST, T. W. CLARK, AND L. RICHARD-SON. 1986. Paleobiology, Biogeography, and Systematicsof the Black-footed Ferret, Mustela nigripes (Audubonand Bachman) 1851. Great Basin Naturalist Memoirs, 8,11-62.

Apa, A. D., D. W. URESK, AND R. L. LINDER. 1990. Black-tailed Prairie Dog Populations One Year After Treat-ment with Rodenticides. Great Basin Naturalist, 50,107-113.

BEVERS, M., J. HO F, B. KENT, AND M. G. RAPHAEL. 1995.Sustainable Forest Management for Optimizing Multi-species Wildlife Habitat: A Coastal Douglas-fir Example.Natural Resource Modeling, 9, l-23.

Bigg ins , D. E., B. J. MILLER, L. R. HANEBURY, B. O a k l e a f ,A. H. FARMER, R. CRETE, AND A. DOOD. 1993. A Tech-nique for Evaluating Black-footed Ferret Habitat. U. S.Fish and Wildlife Service Biological Report, 93, 13, 73-88.

Bigg ins , D. E., M. H. SCHROEDER, S. C. FORREST, AND L.RICHARDSON. 1986. Activity of Radio-tagged Black-footed Ferrets. Great Basin Naturalist Memoirs, 8,135-140.

C i n c o t t a , R. P. 1985. Habitat and Dispersal of Black-tailedPrairie Dogs in the Badlands National Park. UnpublishedM.S. Thesis, Colorado State University, Fort Collins, CO.

506 / BEVERS, HO F, URESK AND SCHENBECK

C i n c o t t a , R. P., D. W. URESK AND R. M. HA N S E N. 1987.Demography of Black-tailed Prairie Dog Populations Re-occupying Sites Treated with Rodenticide. Great BasinNaturalist, 47, 339 -343.

C i n c o t t a , R. P., D. W. URESK, AND R. M. HANSEN. 1988. AStatistical Model of Expansion in a Colony of Black-tailed Prairie Dogs. In Eighth Great Plains Wildlife Dam-age Control Workshop Proceedings. D. W. Uresk, G. L.Schenbeck and R. Cefkin (tech. coords.), Rapid City,South Dakota. Rocky Mountain Forest and Range Ex-periment Station General Technical Report RM-154,30-33.

CLARK, T. W. 1989. Conservation Biology of the Black-footedFerret, Mustela nigripes. Wildlife Preservation Trust Spe-cial Scientific Report No. 3.

FORREST, S. C., T. W. CLARK, L. RICHARDSON , AND T. M.CAMPBELL III. 1985. Black-footed Ferret Habitat: SomeManagement and Reintroduction Considerations. Wyo-ming Bureau of Land Management Technical BulletinNo. 2.

GURNEY, W. S. C. AND R. M. NISBET. 1975. The Regulation ofInhomogeneous Populations. J. Theoret. Biology, 52,441-457.

Hanski , I. 1994. A Practical Model of Metapopulation Dy-namics. J. Animal Ecology, 63, 151-162.

HaRRIS, R. B., T. W. CLARK, AND M. L. SHAFFER. 1989. Ex-tinction Probabilities for Isolated Black-footed FerretPopulations. In Conservation Biology and the Black-footedFerret. U. S. Seal, E. T. Thorne, M. A. Bogan and S. H.Anderson (eds.), Yale University Press, New Haven andLondon.

HENDERSON, F. R., P. F. SPRINGER, AND R. ADRIAN. 1969. TheBlack-footed Ferret in South Dakota. South Dakota Dept.of Game, Fish and Parks Technical Bulletin No. 4,Pierre, SD.

H i l l m a n , C. N., R. L. LINDER, AND R. B. D AHLGREN. 1979.Prairie Dog Distributions in Areas Inhabited by Black-footed Ferrets. American Midland Naturalist, 102,185-187.

HOF, J., M. BEVERS, L. JOYCE, AND B. KENT. 1994. An IntegerProgramming Approach for Spatially and TemporallyOptimizing Wildlife Populations. Forest Sci. 40, 177-191.

HOOGLAND, J. L., D. K. ANGELL, J. G. D A L E Y, AND M. C.RADCLIFFE . 1988. Demography and Population Dynamicsof Prairie Dogs. In Eighth Great Plains Wildlife DamageControl Workshop Proceedings. D. W. Uresk, G. L. Schen-beck and R. Cefkin (tech. coords.), Rapid City, SouthDakota. Rocky Mountain Forest and Range ExperimentStation General Technical Report RM-154.

HOUSTON, B. R., T. W. CLARK, AND S. C. M i n t a . 1986. HabitatSuitability Index Model for the Black-footed Ferret: AMethod to Locate Transplant Sites. Great Basin Natural-ist Memoirs, 8, 99-l 14.

KIERSTEAD, H. AND L. B. SLOBODKIN . 1953. The Size of WaterMasses Containing Plankton Blooms. J. Marine Res. 12,141-147.

KNOWLES, C. J. 1985. Population Recovery of Black-tailedPrairie Dogs Following Control with Zinc Phosphide. J.Range Mgmt. 39, 249-251.

LINDER, R. L., R. B. DAHLGREN, AND C. N. H i l l m a n , 1972.Black-footed Ferret-Prairie Dog Interrelationships. InSymposium on Rare and Endangered Wildlife of the South-

western United States. September, Albuquerque, NM,New Mexico Dept. of Game and Fish.

MACARTHUR, R. H. AND E. 0. WILSON. 1967. The Theory ofIsland Biogeography. Princeton University Press, NewJersey.

MCDONALD, P. M. 1995. Black-footed Ferret Monitoring, Buf-falo Gap National Grassland, Winter 1994. UnpublishedReport, Nebraska National Forest, Chadron.

M ILLER, B. J., G. E. M e n k e n s , AND S. H. ANDERSON. 1988. AField Habitat Model for Black-footed Ferrets. In EighthGreat Plains Wildlife Damage Control Workshop Proceed-ings. D. W. Uresk, G. L. Schenbeck and R. Cefkin (tech.coords.), Rapid City, South Dakota. Rocky MountainForest and Range Experiment Station General TechnicalReport RM-154.

M i n t a , S. AND T. W. CLARK. 1989. Habitat Suitability Analysisof Potential Translocation Sites for Black-footed Ferretsin North-central Montana. In The Prairie Dog Ecosystem:Managing for Biological Diversity. T. W. Clark, D. Hinck-ley, and T. Rich (eds.), Montana Bureau of Land Man-agement, Billings, MT, 29 - 46.

O a k l e a f , B., B. LUCE, AND E. T. THORNE. 1992. Evaluation ofBlack-footed Ferret Reintroduction in Shirley Basin, Wy-oming. In 1991 Annual Completion Report. B. Oakleaf, B.Luce, E. T. Thorne and S. Torbit (eds.), Wyoming Gameand Fish Department, Cheyenne, WY, 196-240.

O a k l e a f , B., B. LUCE, AND E. T. THORNE. 1993. An Evalua-tion of Black-footed Ferret Reintroduction in Shirley Ba-sin, Wyoming. In 1992 Annual Completion Report. B.Oakleaf, B. Luce, E. T. Thorne and B. Williams (eds.),Wyoming Game and Fish Department, Cheyenne, WY,220 -234.

OKUBO, A. 1980. Diffusion and Ecological Problems: Mathe-matical Models. Springer-Verlag, New York.

PLUMB, G. E., P. M. MCDONALD, AND D. SEARLS. 1994. Black-footed Ferret Reintroduction in South Dakota: ProjectDescription and 1994 Protocol. Unpublished Document,Badlands National Park.

RICHARDSON, L., T. W. CLARK, S. C. FORREST, AND T. M.CAMPBELL III. 1987. Winter Ecology of Black-footed Fer-rets (Mustela nigripes) at Meeteetse, Wyoming. AmericanMidland Naturalist, 117, 225-239.

ROEMER, D. M. AND S. C. FORREST . 1996. Prairie Dog Poison-ing in the Northern Great Plains: An Analysis of Pro-grams and Policies. Environmental Mgmt. 20, 349-359.

SCHENBECK, G. L. AND R. J. MYHRE. 1986. Aerial Photographyfor Assessment of Black-tailed Prairie Dog Managementon the Buffalo Gap National Grassland, South Dakota.USDA Forest Service FPM-MAG Report No. 86-7.

SEAL, U. S. 1989. Introduction. In Conservation Biology andthe Black-footed Ferret. Yale University Press, New Ha-ven and London.

SKELLAM, J. G. 1951. Random Dispersal in Theoretical Popu-lations. Biometrika, 38, 196-218.

THORNE, E. T. AND D. W. BELITSKY. 1989. Captive Propaga-tion and the Current Status of Free-ranging Black-footedFerrets in Wyoming. In Conservation Biology and theBlack-footed Ferret. Yale University Press, New Havenand London, 223-234.

URESK, D. W., J. G. M a c C r a c k e n , AND A. J. BJUGSTAD. 1981.Prairie Dog Density and Cattle Grazing Relationships. In

B EVERS, HO F, Uresk AND SCHENBECK / 507

Fifth Great Plains Wildlife Damage Control Workshop U. S. FISH AND WILDLIFE SERVICE, NATIONAL PARK SERVICE

Proc. Lincoln, NB, 199-201. AND USDA FOREST SERVICE. 1994. Black-footed FerretURESK, D. W. AND G. L. SCHENBECK. 1987. Effect of Zinc Reintroduction, Conata Basin/Badlands, South Dakota.

Phosphide Rodenticide on Prairie Dog Colony Expan- Final Environmental Impact Statement.sion as Determined from Aerial Photography. Prairie WALTERS, C. 1986. Adaptive Management of Renewable Re-Naturalist, 19, 57-61. sources. Macmillan Publishing Co., New York.