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Optimal Foraging and Predator-Prey Dynamics, II Theoretical Population Biology 55, 111126 (1999) Article ID tpbi.1998.1399, available online at http:ÂÂwww.idealibrary.com on Optimal Foraging and PredatorPrey Dynamics, II Vlastimil Kr ivan and Asim Sikder Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, and Faculty of Biological Sciences USB, Branis ovska 31, 370 05 C8 eske Bude jovice, Czech Republic E-mail: krivanÄentu.cas.cz Received May 11, 1997 In this paper we consider one-predatortwo-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predatorprey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press Key Wordsy optimal foraging; population dynamics; apparent competition; persistence; permanence. INTRODUCTION profitable prey type is more gradual than predicted by optimal foraging theory. In Kr ivan (1996) a two-preyone-predator population Kr ivan (1996) showed that optimal foraging leads dynamic model with optimal predator foraging behavior naturally to a more general class of population dynami- was considered. This model assumes that predators cal systems which are described by differential inclusions forage according to optimal foraging theory (Charnov, (i.e., by differential equations with multivalued right- 1976; Stephens and Krebs, 1986), which predicts that the hand sides), and to the emergence of partial preferences more profitable prey type is always included in the for the alternative prey type as a consequence of popula- predator diet while the less profitable (i.e., alternative) tion dynamics. The multivaluedness in the model des- prey type is included with probability one only if the den- cription is due to the non-uniqueness of optimal foraging sity of the more profitable prey type falls below a critical strategy when the more profitable prey type reaches the threshold. Therefore, the optimal foraging model does critical threshold. It was shown that the interplay among not predict the emergence of partial preferences for the behavioral ecology and population dynamics is twofold: alternative prey type which is either completely included optimal foraging influences population dynamics, which, or excluded from predators' diet. The predictions of in turn, affects optimal prey diet. Analysis given in optimal foraging theory were compared with several Kr ivan (1996) focused mainly on local stability of the experimental and field studies in Stephens and Krebs ecological equilibrium and analytical results given there (1986) (see also Richardson and Verbeeck (1992)). These were only for a special case in which both prey types had studies support the idea that the diet choice is based on the same intrinsic per capita growth rate. The reason for food profitability although the inclusion of the less this limitation was due to the fact that even local stability 111 0040-5809Â99 K30.00 Copyright ] 1999 by Academic Press All rights of reproduction in any form reserved. 112 Kr @van and Sikder analysis for the general case gives very complex expres- predators follow optimal diet choice and predatorprey sions which cannot be efficiently analyzed. On the population dynamics is of the RosenzweigMacArthur numerical evidence it was conceived there that optimal type. foraging may lead to permanence in the one-predator two-prey model; i.e., all three populations can coexist indefinitely. Optimal foraging can also reduce amplitude POPULATION DYNAMICS of fluctuations in population densities when compared with generalist predator behavior. In this part we survey the model introduced in Kr ivan In this paper we study analytically the effect of optimal (1996) where more details can be found. We consider foraging on permanence of the one-predatortwo-prey predators foraging on two prey types. Predator density is system introduced in Kr ivan (1996). We show that the denoted by x , and prey densities are x and x , respec- conditions for permanence are a great deal simpler and 3 1 2 tively. Population dynamics is described by a system of more efficiently tractable than those for local asymptotic differential equations stability. To conclude that all populations can coexist, permanence theory (Hofbauer and Sigmund, 1984; x1 u1 *1 x1 x3 Butler and Waltman, 1986; Hutson and Schmitt, 1992) x$1 =a1 x1 1& & does not require consideration of the complicated \ K1 + 1+u1 h1 *1 x1+u2 h2 *2 x2 behavior of interior (i.e., all populations present) orbits x2 u2 *2 x2 x3 of the model found in Kr ivan (1996). Permanence means x$2=a2 x2 1& & (1) K 1+u h * x +u h * x that there is a lower positive bound such that in a long \ 2 + 1 1 1 1 2 2 2 2 term run all population densities (initially positive) will u1 e1 *1 x1 x3+u2 e2 *2 x2 x3 be above this bound and no population density tends to x$3= &mx3 . 1+u1 h1 *1 x1+u2 h2 *2 x2 infinity. In Butler and Waltman (1986) the results con- cerning permanence are given for continuous dynamical Here u denotes the probability that a predator will systems, i.e., for systems in which trajectories depend i attack the prey type i, * is the search rate of a predator continuously on initial data. Since the system studied in i for the ith prey type, e is the expected net energy gained Kr ivan (1996) is described by differential inclusions i from the ith prey type, and h is the expected handling which define a continuous dynamical system, we can use i time spent with the ith prey type. In what follows we the results of Butler and Waltman (1986) to study its per- assume that either u and u are fixed if predators show manence. 1 2 fixed preferences for their prey or they are chosen accord- We consider the effects of three types of predator ing to the optimal diet model. Maximization of the net foraging behaviors: (i) predators specialize on the more energy intake rate which is used as the fitness measure in profitable prey type only, (ii) predators are generalists optimal foraging theory (Stephens and Krebs, 1986; and they feed on both prey types, and (iii) predators Kr ivan, 1996) is equivalent to maximization of the forage adaptively following rules of optimal foraging instantaneous per capita predator growth rate x$ Âx .We theory. We derive and compare permanence conditions 3 3 remark that model (1) does not consider any direct com- for the system incorporating the above three predator petition between the two prey populations, but the two foraging behavior types. We show that the region of the prey types are in apparent competition through the parameter space for which permanence holds for adap- shared predation (Holt, 1977). Of course, this apparent tive foragers is larger than that for non-adaptive competition appears only provided that both prey types generalist foragers, but smaller than that for specialist are included in predators' diet. foragers which feed only on the more profitable prey Throughout the paper we will assume that the first type. prey type is more profitable for predators than the We do not consider any direct competition among the second alternative prey type, by which we mean that two prey types. Therefore, in the predator absence both prey populations coexist at their equilibrium levels. e e When predators are introduced, predator-mediated 1 > 2 . apparent competition (Holt, 1977) among prey popula- h1 h2 tions may drive on prey type to extinction. For generalist predators the indirect effects of predation on competition If predators follow optimal diet choice, the optimal among the two prey types were studied by Holt (1977). strategy of a predator when encountering a prey depends In this paper we extend this study to the case where on the density of the more profitable prey type which is Optimal Foraging 113 always attacked upon an encounter, i.e., u =1. The x * x x 1 x$ =a x 1& 1 & 1 1 3 alternative prey type is attacked with probability one 1 1 1 \ K1 + 1+h1 *1 x1 (u1=1) if the density of the more profitable prey type is x2 below the critical threshold x$2=a2 x2 1& (3) \ K2 + e2 e1 *1 x1 x3 x*1= . x$3= &mx3 . *1(e1 h2&e2 h1) 1+h1 *1 x1 Note that the equation for x in (3) is independent of the If the first prey type density is above x* then the alter- 2 1 other two equations. native prey type is not attacked upon an encounter In G0 the right-hand side of (1) is not uniquely defined (u =0) since it pays off for predators to search for the 2 since 0u 1. However, it was shown in Kr ivan (1996) more profitable prey type. Following Murdoch and 2 that in the subregion of G0 described by Oaten (1975) we call x*1 the switching density because predators switch their behavior at this prey density. If the a x* first prey type density equals x* then optimal predator 1 1& 1 (1+h * x*) 1 * K 1 1 1 strategy is not uniquely defined by maximization of 1 \ 1 + x$ Âx , i.e, 0u 1. Thus, control u as a function of the a x* 3 3 2 2 <x < 1 1& 1 (1+h * x*+h * x ) more profitable prey type density is a step-like function 3 1 1 1 2 2 2 *1 \ K1 + and (1) which is driven by optimal predator foraging strategy is not a differential equation but a differential trajectories of (1) cannot leave G0.
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