EVOLUTIONARY STABILITY OF IDEAL FREE DISPERSAL STRATEGIES IN PATCHY ENVIRONMENTS ROBERT STEPHEN CANTRELL, CHRIS COSNER, AND YUAN LOU Abstract. A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionary stable. Hastings [26] showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt’s work [38] suggested that among conditional dispersal strategies in a spatially heterogeneous but tem- porally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionary stable. We use continuous- time and discrete-space models to determine when the dispersal strategy with no movement is evolutionary stable and when an ideal dispersal strategy is evolutionary stable, both in a spatially heterogeneous but temporally constant environment. Keywords: Evolution of dispersal; Ideal free distribution; Evolutionary stability; Neigh- borhood invader strategy; Patchy environments AMS Classification: 34D23, 92D25 1. Introduction Dispersal is an essential aspect of ecology. It is important because it affects population interactions, biological invasions, and the geographical distributions of populations, and their response to habitat fragmentation, among other things. The effects and evolution of dispersal have been studied extensively by theoretical ecologists [12]. The problem of understanding the evolution of dispersal strategies in spatially heterogeneous but temporally constant environments has received considerable attention [3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 19, 25, 26, 27, 30, 31, 33, 34, 35, 38, 40]. In this paper we will examine that question in the context of discrete diffusion models. The corresponding problem in the case of spatial and temporal variation was considered in [28, 29]; See also [21, 41, 42, 43] for some recent important progress in this direction, but we will not address that in the current paper. An important distinction among dispersal strategies is whether they are conditional (depending on environmental factors) or unconditional (effectively random, or at least not based on response to the environment). We will see that when dispersal is favored at all, conditional dispersal of a certain type is favored over unconditional dispersal. Our results extend and refine previous work on that topic. We will take the viewpoint of adaptive dynamics [16, 17, 18, 23, 24, 39]. An important idea in adaptive dynamics is the idea of evolutionarily stable strategies (ESS). A strategy is said to be evolutionarily stable if a population using it cannot be invaded by any small population using a different strategy. A related but different idea is that of convergent stable strategies (CSS). A strategy is convergent stable if small changes in nearby strategies are only favored (i.e., able to invade a resident population) if they are closer to the convergent stable strategy than the resident strategy. The key idea is whether Date : October 18, 2011. 1