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The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system.

In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

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