Stability (learning theory)

Physics

Nonlinear system

Mathematics

Domain (mathematical analysis)

Boundary (topology)

Evolutionary biology

Biology

Statistical physics

Dynamics (music)

Machine learning

Mathematical analysis

Function (biology)

Artificial intelligence

Predation

Neumann boundary condition

Exponential stability

Acoustics

Ecology

Chaotic

Computer science

Quantum mechanics

Applied mathematics

Bounded function

Lyapunov function

European Journal of Applied Mathematics

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.

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