Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator-prey system with Crowley-Martin functional response

A diffusive predator-prey system with Crowley-Martin functional response is considered. Firstly, the maximal parameter region, where the coexistence equilibrium is stable, is provided, of which the boundary consists of Turing bifurcation curves and Hopf bifurcation curve, and result derived by Shi and Ruan (2015) is improved. Meanwhile, critical conditions for Turing instability are derived, which are necessary and sufficient. Furthermore, considering the degenerated situation where Turing bifurcation and Hopf bifurcation occur simultaneously, conditions for codimension-two Turing-Hopf bifurcation and Turing-Turing bifurcations are given. For Turing-Hopf bifurcation, by analyzing the normal forms truncated to order 3, which are derived by applying normal form method and generic formulas developed by Jiang, An and Shi (2018), it is found that system exhibits spatial, temporal and spatiotemporal patterns, like transient spatially inhomogeneous periodic solutions, as well as tristable phenomena of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. At last, numerical simulations, including transient, bistable and tristable patterns, are illustrated to support our theory results.