Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect

• To have spatial patterns form, small Allee effect admits relatively large range of predator mortality rate.
• Found cross-diffusion is the key mechanism of spatial pattern formation.
• Obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur.
• Weak Allee effect admits supercritical bifurcation, while strong one allows subcritical bifurcation.

In this paper, we first propose a mathematical model for a spatial predator-prey system with Allee effect. And then by using the proposed model, we investigate the Turing instability and the phenomena of pattern formation. We show how cross-diffusion destabilizes the spatially uniform steady state. The method of multiple time scales is employed to derive the amplitude equations, which is the cubic Stuart-Landau equation in the supercritical case and the quintic in the subcritical case. Based on the amplitude equations, we obtain the asymptotic solutions of the model close to the onset of instability.