On several conjectures from evolution of dispersal

Abstract

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17-36 [9]] concerning the dynamics of a diffusion-advection-competition model for two competing species.A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [9]. It was shown in [9] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [9] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [9], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.

Competitive–cooperative models with various diffusion strategies

“…In order to involve movements in spatially distributed systems, the dispersal strategy should be specified. The ideas of spreading over the domain to avoid overcrowding combined with advection towards higher available resources were incorporated in the model in [1,5,6,7,8]. Dispersal design in [6] was based on the notion of the ideal free distribution, i.e.…”

Evolutionary Convergence to Ideal Free Dispersal Strategies and Coexistence

“…That is, for appropriately chosen m, α, β, μ, and ν, where μ < α < β, both (θ α,μ , 0) and (0, θ β,ν ) can be unstable. Similar to Theorem 6 of Averill et al (2011), we have the following result. ln m(x 0 ) < Ω m 2 ln m Ω m 2 .…”

“…In Sect. 4, we establish a result which generalizes previous work of Averill et al (2011). Sections 5-8 are devoted to the proofs of the main results on the convergent stability of the ideal free dispersal strategy.…”

“…The full generality in current form was recently given in Averill et al (2011). In terms of the theory of Adaptive Dynamics (Dieckmann 1997(Dieckmann , 2003Geritz et al 1998Geritz et al , 1996, the strategy P = ln m is evolutionarily stable.…”

“…We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously.…”