Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

• Published in: European journal of applied mathematics Vol. 32; no. 4; pp. 652 - 682
• Main Authors: JIN, HAI-YANG, WANG, ZHI-AN
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.08.2021
• Subjects: Applied mathematics; Boundary conditions; Density; Dynamic stability; Dynamical systems; Motility; Motion stability; Nonlinear systems; Parameters; Predator-prey simulation; Predators; Steady state
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792520000248

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.