Optimization of species survival for logistic models with non-local dispersal

To describe the effects of spatial dispersal strategies and heterogeneous environments on the dynamics of species, we study a logistic model with non-local dispersals for single species. The aim of this paper is to study the optimization of species survival and the investigations are carried out from two aspects: optimization of survival chances and maximization of total population of species. W.l.o.g., assume that the spatial distribution of resources m∈L∞(Ω)m∈L∞(Ω) satisfies −1≤m(x)≤1and1|Ω|∫Ωm(x)dx≤α,α∈(−1,1). Our main results indicate that m=χE−χΩ∖Em=χE−χΩ∖E, where E⊂ΩE⊂Ω is measurable with |E|/|Ω|=(1+α)/2|E|/|Ω|=(1+α)/2 and χEχE is the characteristic function of EE, is optimal for both the survival chances and total population of species respectively.