Abstract
We studied a delayed predator–prey model with diffusion and anti-predator behavior. Assume that additional food is provided for predator population. Then the stability of the positive equilibrium is considered. The existence of Hopf bifurcation is also discussed based on the Hopf bifurcation theory. The property of Hopf bifurcation is derived through the theory of center manifold and normal form method. Finally, we analyze the effect of time delay on the model through numerical simulations.
Highlights
The interaction between predator and prey is ordinary and widespread in nature, it could affect the ecological balance
We incorporated time delay and diffusion on the model (1), and studied the dynamics in a delayed diffusive system with anti-predator and additional food provided for a predator
By using time delay as a parameter, we mainly studied the local stability of coexisting equilibrium, the existence of Hopf bifurcation induced by delay, and the property of Hopf bifurcation by the theory of the center manifold and normal form method
Summary
The interaction between predator and prey is ordinary and widespread in nature, it could affect the ecological balance. Bentout studied a diffusive predator–prey model in the presence of predator rivalry and prey social behavior. They mainly studied the stability and Hopf bifurcation, and they show the nonhomogeneous periodic solution induced by diffusion. These works show that the diffusion term often causes the Turing pattern, spatial non-uniform periodic oscillation and so on. We mainly study the effect of time delay and diffusion term on the model (3), such as delay inducing instability, homogeneous or inhomogeneous bifurcating periodic solutions.
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