Global existence and stabilization in a diffusive predator–prey model with population flux by attractive transition

Abstract

The diffusive Lotka–Volterra predator–prey model ut=∇⋅d1∇u+χv2∇(uv)+u(m1−u+av),x∈Ω,t>0,vt=d2Δv+v(m2−bu−v),x∈Ω,t>0,is considered in a bounded domain Ω⊂Rn, n∈{2,3}, under Neumann boundary condition, where d1,d2,m1,χ,a,b are positive constants and m2 is a real constant. The purpose of this paper is to establish global existence and boundedness of classical solutions in the case n=2 and global existence of weak solutions in the case n=3 as well as show long-time stabilization. More precisely, we prove that the solutions (u(⋅,t),v(⋅,t)) converge to the constant steady state (u∗,v∗) as t→∞, where u∗,v∗ solves u∗(m1−u∗+av∗)=v∗(m2−bu∗−v∗)=0 with u∗>0 (covering both coexistence as well as prey-extinction cases).