On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations

Abstract

Two competing populations in spatially heterogeneous but temporarily constant environment are investigated: one is subject to regular movements to lower density areas (random diffusion) while the dispersal of the other is in the direction of the highest per capita available resources (carrying capacity driven diffusion). The growth of both species is subject to the same general growth law which involves Gilpin-Ayala, Gompertz and some other equations as particular cases. The growth rate, carrying capacity and dispersal rate are the same for both population types, the only difference is the dispersal strategy. The main result of the paper is that the two species cannot coexist (unless the environment is spatially homogeneous), and the carrying capacity driven diffusion strategy is evolutionarily stable in the sense that the species adopting this strategy cannot be invaded by randomly diffusing population. Moreover, once the invasive species inhabits some open nonempty domain, it would spread over any available area bringing the native species diffusing randomly to extinction. One of the important technical results used in the proofs can be interpreted in the form that the limit solution of the equation with a regular diffusion leads to lower total population fitness than the ideal free distribution.

Competitive–cooperative models with various diffusion strategies

“…The analogues of the following statements for less general diffusion types were obtained in [11,12], for completeness we present the proof here.…”

“…For example, in [2,3,10,11,12,13] in the term ∆(u/P ) the dispersal strategy P was usually chosen as P ≡ K guaranteeing that K is a (globally stable) positive solution of the equation. However, if P (x) ≡ K(x)/h(x), where h is any harmonic function on Ω, K is still a solution.…”

“…If all equilibrium solutions but one are unstable, we would be able to conclude that the remaining equilibrium is globally asymptotically stable. We start with the trivial equilibrium, and, similarly to [11,12], verify the following result. Lemma 1.…”

“…In the particular case when P ≡ K (or P is proportional to K) we have the term ∆(u/K), first introduced in [3] and later considered in [10]- [13]. If Q is constant, while a 2 is proportional to 1/K, we have the dispersal type ∇ · ( 1 K ∇v) which was considered in [11,12]. 3.…”

“…Most of the previous research [6,7,8,9,11,12] was focused on evolutionarily stable strategies which provide advantages in a competition. This allowed to answer the question: what parameters or strategies should a spatially distributed population choose so that its habitat cannot be invaded by a species choosing alternative strategy and having other parameters?…”