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) { − Δ u = u ( a 1 − b u − c 1 v u + k 1 ) in Ω , − Δ v = v ( a 2 − c 2 v u + k 2 ) in Ω , u ⩾ 0 , v ⩾ 0 in Ω , u = v = 0 , on ∂ Ω , where Ω ⊂ R N ( N ⩾ 1 ) is a bounded domain with a smooth boundary ∂Ω, the parameters a i , b, c i , k i ( 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.