Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

Abstract EN

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper.

Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed.

It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs.

Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one.

Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function.

The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.