The ideal free distribution as an evolutionarily stable strategy

Abstract

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.

Reaction-diffusion-advection models for the effects and evolution of dispersal

“…( The resulting diffusion term in that case would have the form ∆(u/m(x)).) It was also shown in [26] by means of a local invasibility analysis analogous to those used in [14,43] that the only possible evolutionarily stable strategies in the nonlocal logistic case are those that support an ideal free distribution.…”

Stochastic population growth in spatially heterogeneous environments

“…When movement is towards regions with superior habitat quality, the presence of spatial heterogeneity increases the rate of population growth (Chesson 2000; Schreiber and Lloyd-Smith 2009). The most extreme form of this phenomenon occurs when individuals are able to disperse freely and ideally; that is, they can move instantly to the locations that maximize their per-capita growth rate (Fretwell and Lucas 1970; Cantrell et al 2007). Anthropogenically altered habitats, however, can cause a disassociation between cues used by organisms to assess habitat quality and the actual habitat quality.…”

Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments

“…However, the mathematical analysis for analogous stochastic models with density-dependent feedbacks is largely unexplored. Going beyond single species, the coevolution of patch selection among interacting species has a rich history for spatially heterogeneous, but temporally homogeneous environments (van Baalen and Sabelis, 1993; Křivan, 1997; Schreiber et al, 2000; van Baalen et al, 2001; Schreiber et al, 2002; Cressman et al, 2004; Schreiber and Vejdani, 2006; Cantrell et al, 2007). For example, spatial heterogeneity can select for the evolution of contrary choices in which the prey prefers low quality patches to escape the predator and the predator prefers high quality patches to capture higher quality food items (Fox and Eisenbach, 1992; Schreiber et al, 2000).…”

“…For example, at equilibrium, sedentary populations achieve an ideal-free distribution provided, paradoxically, the populations initially occupied all habitat patches. While many early studies asserted that the ideal free distribution is an evolutionarily stable strategy (ESS) (Fretwell and Lucas, 1969; van Baalen and Sabelis, 1993; Schreiber et al, 2000), only recent advanced nonlinear analyses fully verified this assertion (Cressman et al, 2004; Cressman and Křivan, 2006, 2010; Cantrell et al, 2007, 2010, 2012). …”