Stability (learning theory)

Physics

Nonlinear system

Instability

Bifurcation

Mathematics

Mechanics

Diffusion

Biology

Statistical physics

Distribution (mathematics)

Machine learning

Coupling (piping)

Mathematical analysis

Hopf bifurcation

Pattern formation

Metallurgy

Materials science

Spatiotemporal pattern

Computer science

Quantum mechanics

Genetics

Thermodynamics

Journal of Biological Systems

Spatial patterns through diffusion-driven instability are stationary structures that appear spontaneously upon breaking the symmetry of the spatial domain, which results only from the coupling between the reaction and the diffusion processes. This paper is concerned with a modified Leslie–Gower-type model with cross-diffusion and indirect predation effect. We first prove the global existence, non-negativity and uniform boundedness for the considered model. Then the linear stability analysis shows that the cross-diffusion is the key mechanism of spatiotemporal pattern formation. Amplitude equations are derived near Turing bifurcation point under nonlinear cross-diffusion to interpret pattern selection among spot pattern, stripe pattern and the mixture of spot and stripe patterns, which reflects the species’s spatially inhomogeneous distribution, and it is also found that the fear factor has great influence on spatially inhomogeneous distribution of the two species under certain cross-diffusivity, that is, high level of fear can induce striped inhomogeneous distribution, low level of fear can induce spotted inhomogeneous distribution, and the intermediate level of fear can induce the mixture of spotted and striped inhomogeneous distribution. Finally, numerical simulations illustrate the effectiveness of all theoretical results.

CROSS-DIFFUSION-DRIVEN PATTERN FORMATION AND SELECTION IN A MODIFIED LESLIE–GOWER PREDATOR–PREY MODEL WITH FEAR EFFECT