Bifurcations in the diffusive Bazykin model

Abstract

Dangerous tipping points and catastrophic transitions in ecosystems have recently been popular for detecting early warning signals in ecology. B-tipping is induced by bifurcation such as spatial pattern formation resulting from Turing instability. As one of the most important models in predator-prey interactions, we extend the Bazykin model to incorporate diffusive movement under homogeneous Neumann boundary conditions. For the local model, we provide some preliminary analysis on stability and Hopf bifurcation. For the reaction-diffusion model, we first improve some sufficient conditions for the local and global stability of a semi-trivial constant steady state or a unique positive constant steady state in Du and Lou (2001) [11]. Next we obtain the sufficient and necessary conditions for Turing instability, show the existence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation, Turing-Hopf bifurcation and Turing-Turing-Hopf bifurcation, and the nonexistence of triple-Turing bifurcation. Our results reveal that the model can exhibit complex spatial, temporal and spatiotemporal patterns, including complex regime shifts and critical transitions at bifurcation points, transient states (spatially inhomogeneous periodic solutions), tristability (a pair of non-constant steady states and a spatially homogeneous periodic solution), heteroclinic orbits (connecting a spatially inhomogeneous periodic solution to a non-constant steady state or a spatially homogeneous periodic solution, connecting a spatially homogeneous periodic solution to non-constant steady states and vice versa). Finally, numerical simulations illustrate complex dynamics and verify our theoretical results.