On a competitive system with ideal free dispersal

In this article we study the long term behavior of the competitive system{∂u∂t=∇⋅[α(x)∇um]+u(m(x)−u−bv)inΩ,t>0,∂u∂t=∇⋅[β(x)∇v]+v(m(x)−cu−v)inΩ,t>0,∇um⋅nˆ=∇v⋅nˆ=0on∂Ω,t>0, which supports for the first species an ideal free distribution, that is a positive steady state which matches the per-capita growth rate. Previous results have stated that when b=c=1 the ideal free distribution is an evolutionarily stable and neighborhood invader strategy, that is the species with density v always goes extinct. Thus, of particular interest will be to study the interplay between the inter-specific competition coefficients b,c and the diffusion coefficients α(x) and β(x) on the critical values for stability of semi-trivial steady states, and the structure of bifurcation branches of positive equilibria arising from these equilibria. We will also show that under certain regimes the system sustains multiple positive steady states.