USE OF SELECTION INDICES TO MODEL THE FUNCTIONAL RESPONSE OF PREDATORS

Transcription

1 Notes Ecology, 84(6), 00, pp by the Ecological Society of America USE OF SELECTION INDICES TO MODEL THE FUNCTIONAL RESPONSE OF PREDATORS DAMIEN O. JOLY 1, AND BRENT R. PATTERSON 1 Wisconsin Cooperative Wildlife Research Unit, Department of Wildlife Ecology, University of Wisconsin, 18 Russell Labs, 160 Linden Drive, Madison, Wisconsin 5706 USA Ontario Ministry of Natural Resources, Wildlife Research and Development Section, 00 Water Street, rd Floor N., Peterborough, Ontario, Canada K9J 8M5 Abstract. The functional response of a predator to changing prey density is an important determinant of stability of predator prey systems. We show how Manly s selection indices can be used to distinguish between hyperbolic and sigmoidal models of a predator functional response to primary prey density in the presence of alternative prey. Specifically, an inverse relationship between prey density and preference for that prey results in a hyperbolic functional response while a positive relationship can yield either a hyperbolic or sigmoidal functional response, depending on the form and relative magnitudes of the density-dependent preference model, attack rate, and handling time. As an example, we examine wolf (Canis lupus) functional response to moose (Alces alces) density in the presence of caribou (Rangifer tarandus). The use of selection indices to evaluate the form of the functional response has significant advantages over previous attempts to fit Holling s functional response curves to killing-rate data directly, including increased sensitivity, use of relatively easily collected data, and consideration of other explanatory factors (e.g., weather, seasons, productivity). Key words: Alces alces; Canis lupus; density dependence; disc euation; functional response; predation; prey, multiple; prey selection; Rangifer tarandus; selection indices; wolf prey preference. INTRODUCTION An important component of predicting the regulatory role of a predator is describing the relationship between prey density and number of prey killed per predator in a given unit of time (i.e., the functional response, Holling 1959a, b). Typically, research on this relationship focusses on whether a type II (hyperbolic) or type III (sigmoidal) curve best describes the functional response (e.g., Trexler et al. 1988, Marshal and Boutin 1999), as the combination of functional and numerical responses of a predator to prey density determines the potential for predator regulation of prey (reviewed by Messier [1995]). However, the available data are often insufficient to determine which curve best fits killingrate data for large-mammal predator prey systems. For example, Marshal and Boutin (1999) convincingly demonstrated that killing-rate data would only be sufficient to detect a sigmoidal functional response when the curve was extreme in its curvature. Manuscript received 1 May 00; revised 10 October 00; accepted 6 November 00; final version received 4 December 00. Corresponding Editor: J. M. Ver Hoef. dojoly@wisc.edu A common explanation for a sigmoidal functional response is prey switching (Murdoch and Oaten 1975). For example, a density-dependent preference for a particular prey type would result in a sigmoidal functional response (e.g., Hassell 1978, Chesson 198). Herein we derive a functional-response model that can generate either a hyperbolic or a sigmoidal functional response, depending on how a predator s preference for prey changes with prey density. A hyperbolic functional response is generated if preference is a negative function of prey density, whereas a positive function yields either a hyperbolic or a sigmoidal response depending on the strength of this density relationship. We illustrate the approach by testing for a density-dependent relationship between moose (Alces alces L.) density and wolf (Canis lupus L.) preference for moose over caribou (Rangifer tarandus L.), assuming that moose are the primary prey (Messier 1994, Hayes et al. 000). We also discuss implications of linking the functional response to prey preference for the stability of predator prey systems. DENSITY-DEPENDENT PREY SELECTION A simple index of prey preference is given by Manly et al. (197) and Chesson (1978): 165

2 166 NOTES Ecology, Vol. 84, No. 6 r i/ni i (1) r /n j j1 where r i and n i represent proportional use and availability, respectively, of the ith prey species; i 1,,...,. The selection index euals 1 in the absence of prey selection. For, 1 1. Optimal foraging theory predicts that when alternative prey have differing habitats, a predator will always select to forage for primary prey unless the primary prey is below a threshold density (Oksanen et al. 001). However, sampling for alternative prey should result in a monotonic rather than stepwise transition in prey preference as primary prey density increases (Oksanen et al. 001). The effects of primary-prey density (x p ), alternative prey density (x alt ), as well as other correlates (y) on primary-prey preference ( p ) can be examined using a sigmoidal logit model: p j k p p alt alt l l ln x x y. () 1 l1 p where the subscripts p and alt represent primary and alternative prey species (here and throughout the manuscript), and k other correlates are included. In the example below we use standard model-selection techniues to examine the effect of moose and caribou densities on selection for moose by wolves. Note that although multiple prey species can be included in E., the euations we describe below become intractable beyond three species because preference for each of greater-than-three species must be modeled using a multinomial logit-transform. Therefore, we present this method as being most useful when examining preference and functional responses for a target primary prey species as a function of its density (x p ), and the density of alternative prey (x alt ). A hyperbolic functional response has a declining slope over all densities, whereas a sigmoidal functional response has an inflection point at some prey density. Therefore the two models can be distinguished by determining whether the curve is convex (hyperbolic) or concave (sigmoidal) at zero prey density using the second-derivative test (i.e., a positive second derivative indicates concavity). Following Hassell (1978) we link the hyperbolic and sigmoidal functional responses by assuming that the attack rate (a) of Holling s (1959a, b) disc euation is a function of prey density. However, the asymptotic attack-rate function of Hassell (1978) does not allow for attack rate to be a negative function of prey density and so lacks generality. Chesson (198) showed that the attack rate, a, was related to Manly and colleagues (197) prey selection index ( i ) as: r i/ni ai i () r /n a j1 j j j j1 for i 1... prey species. For mathematical convenience we assume that the predator has an arbitrary total attack rate (a t ) that is the sum of prey-specific attack rates (a t a p a alt, similar to the maximum search efficiency parameter of Oksanen et al. [001]). Assuming that attack rate can be divided among prey assumes that the predator does not simultaneously search for multiple prey. We defined a multiprey functional response as atpxp Ep (4) 1 a hx a (1 )h x t p p p t p alt alt where E p is the number of primary prey killed per unit time per predator, a t is the total attack rate, and, h, and x refer to preference index, handling time, and density of each prey, respectively. This formulation is structurally euivalent to that proposed by Oksanen et al. (001: E. ) and Osenberg and Mittlebach (). To predict the functional response of a predator to primary prey density, p is then defined as a function of prey density. Substituting E. into E. 4, representing preference for alternative prey as (1 p ), differentiating with respect to density of primary prey twice, and setting primary prey density to zero yields d (E p) dx p xp0 alt (a h e x ah x )a e alt t p alt p p t xalt. altxalt (1 e ah x ) (5) A sigmoidal curve results when E. 5 is positive (and thus concave); therefore the criterion for a sigmoidal functional response is p, where altx ahe alt t p. (6) 1 ah x Thus, the slope ( p ) of the relationship between primary prey density and logit-transformed preference for primary prey dictates the potential for a sigmoidal functional response. Specifically, if p then the functional response is hyperbolic. We used this criterion to demonstrate that coyotes (Canis latrans) show a hyperbolic functional response to white-tailed deer (Odocoileus virginianus) density (B. R. Patterson, D. O. Joly, and F. Messier, unpublished manuscript). EXAMPLE MOOSE AND CARIBOU Considerable debate has surrounded the form of the functional response of wolves to changes in density of

3 June 00 NOTES 167 Data on wolf prey from published studies that provided use (number of each prey type killed) and availability (prey density) in Alaska, USA, and the Yukon Territory, Canada, where wolves killed both caribou and moose. TABLE 1. Source Study area Year Density (no./km ) Use Preference, Hayes et al. (000) Ballard et al. (1997) Northwest Alaska Dale et al. (1995) Mech et al. (1995), Adams et al. (1995) Walker Lake, northern Alaska Iniakuk, northern Alaska Unakserak, northern Alaska Pingaluk, northern Alaska Denali National Park, Alaska Denali National Park, Alaska Caribou density estimated for 1990 and using population growth rates in Hayes et al. (000). Caribou and moose density estimates from Adams et al. (1996). Caribou population density estimated using May census of females, assuming 1:1 sex ratio moose, their primary prey (e.g., Messier 1994, 1995, Eberhardt 1997, 1998, Eberhardt and Peterson 1999, Marshal and Boutin 1999, Eberhardt 000, Hayes and Harestad 000, Messier and Joly 000). Data on wolf prey use (number of each prey type killed) and availability (prey density) were found in the literature (Table 1), allowing us to illustrate the use of our method to determine whether a sigmoidal or hyperbolic functional response is most likely for wolf moose systems. We assessed whether the criteria for a sigmoidal functional response (E. 6) was met by estimating the strength of the density dependence in wolf preference for moose, p, and comparing it to the value of determined by Monte Carlo simulation. The parameters, p, and alt were estimated by fitting E. to wolf prey selection data (Table 1) using a generalized linear model (R, version 1.5 [001]; Ihaka and Gentleman 1996; parameter estimates with 1 SE in parentheses 0.68 (0.84), p 5.79 (.16), alt 0.16 (0.7), Table ). These parameter estimates were model averaged using Akaike weights (Table ; see Burnham and Anderson [1998: ]). Handling time for moose (h p [mean 1 SE]) and total attack rate (a t ) were estimated by fitting Holling s (1959a, b) disc euation to functional response data in single prey (moose) and wolf systems in Messier (1994) and Hayes and Harestad (000). Handling time for caribou (h c ) was estimated by fitting Holling s disc euation to killing-rate data in Dale et al. (1994). Parameter estimates were determined using the sums of suares method (Hilborn and Mangel 1997: ) with the nlm function in R, version 1.5 (001) (Ihaka and Gentleman 1996). Standard errors were estimated from the inverse of the Hessian matrix. We assumed that in the absence of alternative ungulate prey, wolves would devote all their hunting pressure to moose; thus the attack rate estimated from these data may approximate the total attack rate, a t (i.e., in the presence of an alternative prey, the total attack rate would be divided among the prey in proportion to preference). Monte Carlo simulation was used to estimate the mean and 95% confidence intervals for. The simulation was repeated assuming the 5th, 50th, and 75th percentile caribou densities used in this study (0.158, 0., and 0.8 caribou/km, respectively; Table 1). Ninety-five percent confidence intervals (CI) for were estimated as the.5 and 97.5 percentiles from a distribution of values of generated by picking a TABLE. Models of wolf preference for moose as a function of moose and caribou densities. Model Parameters R parameters No. 1 moose moose and caribou caribou AIC, score Akaike weight Notes: Model-averaged parameter estimates with 1 SE in parentheses were: intercept () 0.68 (0.84), moose coefficient ( p ) 5.79 (.16), caribou coefficient ( alt ) 0.16 (0.7). AIC c small sample size-corrected Aikaike Information Criteria.

4 168 NOTES Ecology, Vol. 84, No. 6 value for each parameter in from a normal distribution using the parameter mean and standard error. We considered wolves to have a sigmoidal functional response if the upper 95% CI for did not overlap with the lower 95% CI for p estimated from the generalized linear model above. Consistent with predictions by Oksanen et al. (001), we found that wolf preference for moose was best described by a model only incorporating moose density, although a model incorporating caribou density also provided a reasonable compromise between bias and explained variance (Table ). Our estimates for for low (.7, 95% CI ), moderate (.14, 95% CI ), and high caribou densities (.90, 95% CI ) were indistinguishable from our estimate for the regression coefficient for moose density ( p 5.79, 95% CI ). Thus we conclude that although preference for moose may be related to moose density (Table ), in this range of caribou densities, wolves exhibit either a very shallow sigmoidal or a hyperbolic functional response. DISCUSSION Using density-dependent selection indices to link the functional response to prey preference illustrates some important features of this component of predation. Rather than viewing functional responses in a categorical fashion as presented by Holling (1959a, b), density-dependent killing rates can be viewed as a continuum based on the degree to which prey preference is altered by factors including relative prey densities in multiple-prey systems. Further, the form of the functional response is not likely to be a stereotypic behavior for a predator. As demonstrated by E. 6, a predator is more likely to exhibit a sigmoidal functional response at high alternative-prey density or if handling time for alternative prey is high. In contrast, at high primaryprey densities, or high values for total attack rate or intercept of the logit-transformed primary-prey preference index, the likelihood of a sigmoidal functional response is reduced. Our modified functional-response model has implications with respect to the stability of predator prey systems (i.e., the ability of the predator to cause prey extinction). Hassell (1978) assumed that attack rate increased with prey density according to a Michaelis- Menten type model. Neglecting density-dependent reduction in prey population growth (as an extinction model is concerned only with low-density populations) the population rate of change of a prey species can be described by dx p rxp Ew p (7) dt where r is the instantaneous population growth for prey species; x p and w are primary prey and predator densities, respectively; and E p is the functional response of the predator to density of the primary prey. Thus, for extinction of a prey species to occur, r must be exceeded by the instantaneous predation rate (p p E p w/ x p ) at very low prey densities. Incorporating a Michaelis-Menten attack rate (a cx/(d x), Hassell 1978) into the multiprey hyperbolic functional response (Murdoch 197), and taking the limit as prey density approaches zero yields an instantaneous predation rate of zero at zero prey density. Thus predatorinduced extinction is unlikely using Hassell s approach. A similar analysis of the functional response curve presented here (E. 4) yields more general results. As primary prey density declines to zero (i.e., limit of p p as x p approaches zero, where p p E p w/x p and E p is as defined in E. 4) the predicted predation rate is estimated by the following: ae altxalt p t p x w. (8) 1 ah x e alt alt E. 8 indicates that regardless of whether the functional response is hyperbolic or sigmoidal in shape, predators can theoretically cause extinction of primary prey if the instantaneous predation rate exceeds the instantaneous population growth rate (i.e., p p r). Predator density is an important determinant of the instantaneous predation rate predicted by E. 8; thus research on the conseuences of predation for prey persistence (e.g., Sinclair et al. 1998) must consider the numerical response of the predator in addition to its functional response (see review by Messier [1995]). To determine the density relationship of the predation rate on primary prey species at very low densities, we took the limit of the slope of the predation curve as prey density approaches zero, and solved for m p, the coefficient of the density-dependent selection index (e.g., E. ). As prey density approaches zero, predation rate will be density dependent if (in the two-prey case) ahe altxalt t p p, (9) 1 ah x which is the same criterion that determines whether the functional response is hyperbolic or sigmoidal (, E. 6). Thus, in the absence of a numerical response (e.g., Messier 1995), predation rate will be inversely density dependent if the functional response is hyperbolic, and density dependent if it is sigmoidal. We note however, that regardless of the form of the functional response, true regulation of the prey population is unlikely when density dependence is relatively weak. In such cases, relatively minor disturbances by other limiting factors may override the stabilizing effects of predator regulation, in particular due to the potential for p p to exceed r (see previous paragraph). Thus, the common suggestion that hyperbolic and sigmoidal func-

5 June 00 NOTES 169 tional responses are inherently destabilizing and stabilizing respectively, is misleading. The use of Manly et al. s (197) selection indices to distinguish between hyperbolic and sigmoidal functional responses provides a significant advantage over attempts to relate kill rate directly to prey density. We concur with Marshal and Boutin (1999) that this latter direct approach is unlikely to distinguish among the functional-response models. We also concur that determining the relationship between density and predation rate directly, particularly through density manipulations, is the most robust way to determine the regulatory role of predation. However, logistical and financial constraints mean that these types of experiments may not be feasible for large mammals. In contrast the method presented here can be applied with more easily obtained data, and various other factors (e.g., environmental conditions or prey and predator densities) can be examined as correlates of prey preference. ACKNOWLEDGMENTS D. O. Joly s postdoctoral appointment at the University of Wisconsin Madison was funded by the USGS-National Wildlife Health Center and the Wisconsin Department of Natural Resources. We thank T. Oksanen, L. Oksanen, P. Turchin, S. Ferguson, J. Chesson, C. Ribic, M. D. Samuel, and J. P. Marshal for kindly reviewing early versions of this manuscript. Discussions with W. J. Rettie led us to consider the functional response in a prey-selection context. LITERATURE CITED Adams, L. G., B. W. Dale, and L. D. Mech Wolf predation on caribou calves in Denali National Park, Alaska. Pages in L. N. Carbyn, S. H. Fritts, and D. R. Seip, editors. Ecology and conservation of wolves in a changing world. Canadian Circumpolar Institute, Edmonton, Alberta, Canada. Ballard, W. B., L. A. Ayres, P. R. Krausman, D. J. Reed, and S. G. Fancy Ecology of wolves in relation to a migratory caribou herd in northwest Alaska. Wildlife Monographs 15:1 47. Burnham, K. P., and D. R. Anderson Model selection and inference: a practical information-theoretic approach. Springer-Verlag, New York, New York, USA. Chesson, J Measuring preference in selective predation. Ecology 59: Chesson, J The estimation and analysis of preference and its relationship to foraging models. Ecology 64: Dale, B. W., L. G. Adams, and R. T. Bowyer Functional response of wolves preying on barren-ground caribou in a multiple-prey system. Journal of Animal Ecology 6: Dale, B. W., L. G. Adams, and R. T. Bowyer Winter wolf predation in a multiple ungulate prey system, Gates of the Arctic National Park, Alaska. Pages 0 in L. N. Carbyn, S. H. Fritts, and D. R. Seip, editors. Ecology and conservation of wolves in a changing world. Canadian Circumpolar Institute, Edmonton, Alberta, Canada. Eberhardt, L. L Is wolf predation ratio-dependent? Canadian Journal of Zoology 75: Eberhardt, L. L Applying difference euations to wolf predation. Canadian Journal of Zoology 76: Eberhardt, L. L Reply: predator prey ratio dependence and the regulation of moose populations. Canadian Journal of Zoology 78: Eberhardt, L. L., and R. O. Peterson Predicting the wolf prey euilibrium point. Canadian Journal of Zoology 77: Hassell, M. P The dynamics of arthropod predator prey systems. Princeton University Press, Princeton, New Jersey, USA. Hayes, R. D., A. M. Baer, U. Wotschikowsky, and A. S. Harestad Kill rate by wolves in the Yukon. Canadian Journal of Zoology 78: Hayes, R. D., and A. S. Harestad Wolf functional response and regulation of moose in the Yukon. Canadian Journal of Zoology 78: Hilborn, R., and M. Mangel The ecological detective: confronting models with data. Monographs in population biology. Princeton University Press, Princeton, New Jersey, USA. Holling, C. S. 1959a. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Canadian Entomologist 91:9 0. Holling, C. S. 1959b. Some characteristics of simple types of predation and parasitism. Canadian Entomologist 91: Ihaka, R., and R. Gentleman R: a language for data analysis and graphics. Journal of Computation and Graphical Statistics 5: [Program R is available free at URL: Manly, B. F. J., P. Miller, and L. M. Cook Analysis of a selective predation experiment. American Naturalist 106: Marshal, J. P., and S. Boutin Power analysis of wolf moose functional responses. Journal of Wildlife Management 6: Mech, L. D., T. J. Meier, and J. W. Burch Patterns of prey selection by wolves in Denali National Park, Alaska. Pages 1 4 in L. N. Carbyn, S. H. Fritts, and D. R. Seip, editors. Ecology and conservation of wolves in a changing world. Canadian Circumpolar Institute, Edmonton, Alberta, Canada. Messier, F Ungulate population models with predation: a case study with the North American moose. Ecology 75: Messier, F On the functional and numerical responses of wolves to changing prey density. Pages in L. N. Carbyn, S. H. Fritts, and D. R. Seip, editors. Ecology and conservation of wolves in a changing world. Canadian Circumpolar Institute, Edmonton, Alberta, Canada. Messier, F., and D. O. Joly Comment: the regulation of moose density by wolf predation. Canadian Journal of Zoology 78: Murdoch, W. W The functional response of predators. Journal of Applied Ecology 10:5 4. Murdoch, W. W., and A. Oaten Predation and population stability. Advances in Ecological Research 9: 11. Oksanen, T., L. Oksanen, M. Schneider, and M. Aunapuu Regulation, cycles, and stability in northern carnivore herbivore systems: back to first principles. Oikos 94: Osenberg, C. W., and G. G. Mittlebach.. Effects of body size on the predator prey interaction between pumpkinseed sunfish and gastropods. Ecological Monographs 59: Sinclair, A. R. E., R. P. Pech, C. R. Dickman, D. Hik, P. Mahon, and A. E. Newsome Predicting effects of predation on conservation of endangered prey. Conservation Biology 1: Trexler, J. C., C. E. McCulloch, and J. Travis How can the functional response best be determined? Oecologia 76:06 14.