Boundedness of solutions in a predator-prey system with density-dependent motilities and indirect pursuit-evasion interaction

This paper is concerned with the predator prey system with density-dependent motilities and indirect pursuit evasion interaction {u(t) = Delta(d(1)(w)u) + u(lambda - u av), x is an element of Omega, t > 0, v(t) = Delta(d(2)(z)v) + v(mu - v - bu), x is an element of Omega, t > 0, 0 = Delta w - w + v, x is an element of Omega, t > 0, 0 = Delta z - z + u, x is an element of Omega, t > 0 in a smooth bounded domain Omega subset of R-n (n >= 1) with homogeneous Neumann boundary conditions, where the parameters lambda, mu, a and b are positive constants. The global existence and the boundedness of the classical solution are established in two dimensions if the motility functions d(1) (w) and d(2) (z) satisfy the following hypotheses d(1) (w) is an element of C-3 ([0, infinity)), d(1) (w) > 0, d(1)'(w) < 0 for all w >= 0, d(2) (z) is an element of C-3 ([0, infinity)), d(2) (z) > 0, d(2)'(z) > 0 and d(2)'(z) is bounded for all z >= 0. However, further assume that the motility function d(2) (z) fulfills more rigorous conditions d(2)(z) is an element of C-3 ([0, infinity)), 0 < d(2)(z) < eta, d(2)'(z) > 0 and d(2)''(z) < 0 for all z >= 0 with eta > 0. It was also proved that the global solutions are uniformly bounded with respect to time in higher dimensions. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the predator-prey system with density-dependent motilities and indirect pursuit-evasion interaction. Under certain assumptions, the global existence and boundedness of classical solutions are proven in two dimensions, and it is also shown that the global solutions are uniformly bounded with respect to time in higher dimensions.