The spatial memory effect in predator and fear effect in prey are incorporated in a diffusive predator-prey model. We are interested in studying the dynamics generated by the memory effect and fear effect, and mainly study the local stability of coexisting equilibrium, the existence of Hopf bifurcation and the property of Hopf bifurcation. Through the numerical simulations, we show that increasing memory-based diffusion coefficient is not conducive to the stability of the coexisting equilibrium, and the fear effect has both stabilizing and destabilizing effect on the coexisting equilibrium under different parameters.

In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter $\eta$, the memory-based diffusion coefficient parameter d, and memory delay $\tau$ can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities.

In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.

The nonlocal competition in prey and schooling behavior among predators are incorporated in a delayed diffusive predator–prey model. Our main interest is to study the dynamic properties of the model generated by nonlocal competition and delay. We mainly concentrate on the stability and Hopf bifurcation at the coexisting equilibrium. Compared with the model without nonlocal competition, our results suggest that nonlocal competition can affect the stability of the coexisting equilibrium, and induce the stably spatial bifurcating periodic solutions.

In this article, a diffusive predator-prey model with memory effect and predation-driven Allee effect is considered. Through eigenvalue analysis, the local asymptotic stability of positive constant steady-state solutions is analyzed, and it is found that memory delay affects the stability of positive constant steady-state solutions and induces Hopf bifurcation. The properties of Hopf bifurcating periodic solutions have also been analyzed through the central manifold theorem and the normal form method. Finally, our theoretical analysis results were validated through numerical simulations. It was found that both memory delay and predation-driven Allee effect would cause the positive constant steady-state solution of the model to become unstable, accompanied by the emergence of spatially inhomogeneous periodic solutions. Increasing the memory period will cause periodic oscillations in the spatial distribution of the population. In addition, there would also be high-dimensional bifurcation such as Hopf–Hopf bifurcation, making the spatiotemporal changes of the population more complex.