http://www.elsevier.com/locate/ytpbi Theoretical Population Biology 63 (2003) 269–279 Optimal foraging and predator–prey dynamics III Vlastimil Krˇivan  and Jan Eisner Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, Branisˇovska´ 31, 370 05 C ˇ eske´Budeˇjovice, Czech Republic Received 2 June 2001 Abstract In the previous two articles (Theor. Popul. Biol. 49 (1996) 265–290; 55 (1999) 111–126), the population dynamics resulting from a two-prey–one-predator system with adaptive predators was studied. In these articles, predators followed the predictions of optimal foraging theory. Analysis of that system was hindered by the incorporation of the logistic description of prey growth. In particular, because prey self-regulation dependence is a strong stabilizing mechanism, the effects of optimal foraging could not be easily separated from the effects of bottom-up control of prey growth on species coexistence. In this article, we analyze two models. The first model assumes the exponential growth of both prey types while the second model assumes the exponential growth of the preferred prey type and the logistic growth of the alternative prey type. This permits the effect of adaptive foraging on two-prey– predator food webs to be addressed. We show that optimal foraging reduces apparent competition between the two prey types, promotes species coexistence, and leads to multiple attractors. r 2003 Elsevier Science (USA). All rights reserved. Keywords: Optimal foraging theory; Adaptive behavior; Predator–prey population dynamics; Apparent competition; Differential inclusion; Bifurcations 1. Introduction It is well known that when two non-competing species share a common predator, predator-mediated apparent competition (Holt, 1977) often leads to competitive exclusion of one prey population. If predation is the only regulatory mechanism of prey exponential growth and predators are non-adaptive foragers, this will always hold for population dynamics modeled by Lotka–Volterra-type differential equations with either a linear or Holling type II functional response. If prey growth is density dependent then both prey species can survive with predators for an appropriate range of parameters (Holt, 1977). Thus, in classical models of population ecology, prey density dependence (bottom- up regulation) relaxes the strength of apparent competi- tion making indefinite coexistence of both prey species possible. In this article, we consider the effect of adaptive foraging (i.e., top-down regulation) on apparent prey competition. In Krˇivan (1996), a two-prey–one-predator population model with optimal predator foraging behavior was studied in a fine-grained environment (Werner and Hall, 1974; Charnov, 1976; Stephens and Krebs, 1986). That model contained two regulatory mechanisms of prey growth: (i) prey self-regulation modeled by the logistic equation (bottom-up control), and (ii) optimal diet selection by predators (top-down control) modeled by a piece-wise Holling type II functional response which arises as a result of optimal predator diet choice. Analysis of the resulting system was hindered by the logistic prey growth and only partial stability analysis for the equal intrinsic per capita prey growth rates was given (Krˇivan, 1996). The analysis together with computer simulations suggested that optimal diet choice reduces the amplitude of population oscillations without stabilizing the system at an equili- brium. Similar results were obtained by Fryxell and Lundberg (1994, 1997) for systems in which switching was described by a more gradual sigmoidal function. However, Krˇivan (1996) also showed that optimal foraging can be destabilizing for certain parameter values. The stable equilibrium of a predator–prey system where predators behave as specialists was destabilized when predators became optimally foraging generalists  Corresponding author. Fax: +420-38-777-5367. E-mail address: krivan@entu.cas.cz (V. K$ rivan). 0040-5809/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0040-5809(03)00012-1