Global existence and asymptotic behavior of a predator–prey chemotaxis system with inter-species interaction coefficients

Abstract

A fully parabolic predator-prey chemotaxis system with inter-species interaction coefficient{u1t=d1Δu1−χ∇⋅(u1∇v1)+u1(σ1−a1u1+e1u2),x∈Ω,t>0,u2t=d2Δu2+ξ∇⋅(u2∇v2)+u2(σ2−a2u2−e2u1),x∈Ω,t>0,v1t=d3Δv1+α1u2−β1v1,x∈Ω,t>0,v2t=d4Δv2+α2u1−β2v2,x∈Ω,t>0, under the homogeneous Neumann boundary conditions in an open, bounded domain Ω⊂Rn with smooth boundary ∂Ω is examined. The parameters are all positive constants and the initial data (u10,u20,v10,v20) are non negative. With some supplementary conditions imposed on the parameters, it is proved that the above system has a unique globally bounded classical solution for n≥2. Moreover, the convergence of the solution is asserted by constructing a suitable Lyapunov functional. If e2, χ2 and ξ2 are sufficiently small, then the solution of the above system converges to a unique positive equilibrium. If e2 is sufficiently large and χ2 is sufficiently small, then the solution converges to the semi-trivial equilibrium point. Remarkably, the convergence rate is exponential when e2≠σ2a1σ1 and algebraic if e2=σ2a1σ1. Finally, the numerical examples validate the outcomes of asymptotic behavior. The results demonstrate the predominant behavior of the parameters a1 and a2 in the existence and stability.