Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes

In this paper, we investigate the existence, multiplicity and stability of positive solutions to a prey-predator model with modified Leslie-Gower and Holling-type II schemes(P){−Δu=u(a1−bu−c1vu+k1)in Ω,−Δv=v(a2−c2vu+k2)in Ω,u⩾0,v⩾0in Ω,u=v=0,on ∂Ω, where Ω⊂RN (N⩾1) is a bounded domain with a smooth boundary ∂Ω, the parameters ai, b, ci, ki (i=1,2) are positive numbers, u and v are the respective populations of prey and predator. Here, we say (u,v) with u|∂Ω=v|∂Ω=0 is a positive solution of problem (P) if (u,v) is a solution of (P) and u,v>0 in Ω.