Stationary problem of a predator–prey system with nonlinear diffusion effects

In this paper, we are concerned with the positive solution set of the following quasilinear elliptic system {−Δ[(1+αv)u]=u(a−u−cv1+mu),x∈Ω,−Δ[(1+γ1+βu)v]=v(b−v+du1+mu),x∈Ω,u=v=0,x∈∂Ω, where ΩΩ is a bounded smooth domain in RNRN,α,β,γα,β,γ are nonnegative constants, a,c,d,ma,c,d,m are positive constants, bb is a real constant. This elliptic system is the stationary problem of a predator–prey model, in which uu and vv denote the population densities of the prey and predator, respectively. Regarding bb as the bifurcation parameter, the global bifurcation structure of the positive solution set is shown. Then the nonlinear effect of either large αα or ββ on the bifurcation point is deduced. Moreover, as αα is large, we show that the positive solution set is only of one type, whereas, as ββ is large, the positive solution set is of two types. Additionally, in certain circumstances, we also show that which one of the two types can characterize the positive solution set.