Global stability of a quasilinear predator-prey model with indirect pursuit-evasion interaction

This paper deals with a predator-prey model with indirect prey-taxis and predator-taxis {u(t) = del center dot (D-1(u)del u) - chi del center dot (S-1(u)del u) + u(alpha v - a(1) - b(1)u), x is an element of Omega, t > 0, u(t) = del center dot (D-1(v)del v) - xi del center dot (S-2(v)del w) + v(alpha(2) - b(2)v - u), x is an element of Omega, t > 0, 0 = Delta w + beta w - gamma w, x is an element of Omega, t > 0, 0 = Delta z + delta v - rho z, x is an element of Omega, t > 0, undem homogeneous Neumann boundary conditionsoothly bounded domain Omega subset of R-n (n >= where the parameters chi, xi, alpha, beta, gamma, delta, rho, a(1), a(2), b(1), b(2) are positive, D-1(u) and D-2(v) are nonlinear diffusion functions, S-1(u) and S-2(v) are nonlinear sensitivity functions. First, under certain suitable conditions for D-i and S-i with i = 1,2, the system admits a unique globally bounded classical solution, provided that b(1) >= 4 alpha, and b(2) > 0. Additionally, by constructing appropriate Lyapuno-v functionals, we investigate the asymptotic stability of the globally bounded solutions and provide the exact convergence rates based on the different parameter choices: When a(1)b(2) < alpha alpha(2), it is shown that the global bounded solution (u, v, w, z) exponentially converges to (a(2)alpha-a(1)b(2)/alpha+b(2)b(1), a(1)alpha-a(2)b(1)/alpha+b(2)b(1), beta(a(2)alpha-a(1)b(2)/gamma+b(2)b(1), delta(a(1)+-a(2)b(1))/rho(alpha+b(2)b(1))) as t -> infinity; When a(1)b(2) > alpha alpha(2), it is shown that the global bounded solution (u, v, w, z) exponentially converges to (0, a(2)/b(2), 0, delta a(2)/rho b(2)) as t -> infinity; When a(1)b(2) > alpha alpha(2), it is shown that the global bounded solution (u, v, w, z) algebraically converges to (0, a(2)/b(2), 0, delta a(2)/rho b(2)) as t -> infinity.