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

A fully parabolic predator-prey chemotaxis system with inter-species interaction coefficient {u(1t) =d(1)Delta u(1) - chi del center dot (u(1)del v(1))+ u(1)(sigma(1) - a(1)u(1) + e(1)u(2)), x is an element of Omega, t > 0, u(2t) = d(2)Delta u(2) +xi v .(u(2)del v(2)) + u(2)(sigma(2) - a(2)u(2) - e(2)u(1)), x is an element of Omega, t > 0, v(1)t =d(3)Delta v(1) + alpha(1)u(2) -beta(1)v(1,) x is an element of Omega, t > 0, v(2t) = d(4)Delta v(2) + alpha(2)u(1) - beta(2)v(2), x is an element of Omega, t > 0, under the homogeneous Neumann boundary conditions in an open, bounded domain Omega subset of R-n with smooth boundary partial differential partial derivative Omega is examined. The parameters are all positive constants and the initial data (u(10), u(20), v(10), v(20)) 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 e(2), chi(2) and xi(2) are sufficiently small, then the solution of the above system converges to a unique positive equilibrium. If e(2) is sufficiently large and chi(2) is sufficiently small, then the solution converges to the semi-trivial equilibrium point. Remarkably, the convergence rate is exponential when e(2) not equal sigma(2)sigma(1)/alpha(1) and algebraic if e(2) = sigma(2)a(1)/sigma(1). Finally, the numerical examples validate the outcomes of asymptotic behavior. The results demonstrate the predominant behavior of the parameters a(1) and a(2) in the existence and stability. (c) 2023 Elsevier Inc. All rights reserved.