Double Hopf Bifurcation in a Delayed Toxic Plankton System With Chemotaxis

In this study, the complex spatiotemporal dynamical behaviors of a diffusive toxic plankton system, subject to time delay and prey-taxis with a Ricker-type sensitivity function, are systematically investigated. Firstly, an analysis of the existence of Codimension-1 Turing bifurcation, Hopf bifurcation, and Codimension-2 Turing–Hopf bifurcation and double Hopf bifurcation are conducted. In particular, we investigate that the addition of chemotaxis term enhances the spatial heterogeneity of the system, thereby inducing Turing instability. Then, a key contribution of this paper lies in departing from the traditional center manifold approach and employing the multiple-timescale method to derive the amplitude equations near the nonzero-mode double Hopf bifurcation point. Subsequently, based on the derived normal form, we analyze the topological structure of orbital distributions near the double Hopf bifurcation point and identify the corresponding spatiotemporal patterns in the original system. The results show that a high chemotactic sensitivity can lead to spatial heterogeneity in the system. The coupling between chemotaxis and toxin delay can induce spatial complexity, such as stability switching, spatially inhomogeneous periodic oscillations, and spatially inhomogeneous aperiodic oscillations.