Persistence and Bifurcation Analysis of a Plankton Ecosystem with Cross-Diffusion and Double Delays

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.