Global stability and Turing instability deduced by cross-diffusion in a delayed diffusive cooperative species model

In this paper, we investigate the dynamical behavior of a delayed diffusive cooperative species model with cross-diffusion. Firstly, in the case of self-diffusion, we study the persistence properties and global stability of positive equilibrium for this model by constructing Lyapunov function. The obtained results reveal that the delay has no effect on the stability of positive equilibrium for this model. However, cross-diffusion can affect the stability of positive equilibrium of this model. Then, we continue to discuss Turing bifurcation on one-dimensional space and Turing pattern on two-dimensional space, which are deduced by cross-diffusion. For Turing bifurcation, choosing some cross-diffusion rate as bifurcation parameter, this model undergoes Turing bifurcation nearby the positive equilibrium as cross-diffusion rate is across the Turing bifurcation curve, and bifurcates the stable spatially inhomogeneous steady state solutions. For Turing pattern, we find out the Turing region under some conditions successfully. Selecting the different values of two cross-diffusion rates in the Turing region respectively, we carry out some numerical simulations and obtain spots pattern, spots-strip pattern and strip pattern.