Global dynamics of a three-species spatial food chain model

In this paper, we study the following initial-boundary value problem of a three-species spatial food chain model {u(t) = d(1 )delta u + u(1 - u) - b(1)uv, x is an element of omega, t > 0 v(t )= d(2)delta v - & nabla; center dot (xi v & nabla;u) + uv - b(2)vw - theta(1)v, x is an element of omega, t > 0 w(t)= delta w- & nabla; center dot (chi w & nabla;v) + vw - theta(2)w, x is an element of omega, t > 0 in a bounded domain omega subset of R-2 with smooth boundary and homogeneous Neumann boundary conditions, where all parameters are positive constants. By the delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data. Moreover by constructing Lyapunov functionals and using LaSalle's invariance principle, we establish the global stability of the prey-only steady state, semi-coexistence and coexistence steady states. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we study the initial-boundary value problem of a three-species spatial food chain model in a bounded domain. We establish the global existence of classical solutions and prove the global stability of prey-only steady state, semi-coexistence, and coexistence steady states.