Abstract
In this paper, we study the dynamics of a delayed reaction–diffusion predator–prey model with anti-predator behaviour. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the model, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. By utilizing upperlower solution method, the global asymptotic stability of the interior equilibrium is studied. Finally, numerical simulation results are presented to validate the theoretical analysis.
Highlights
The predator–prey model first proposed by Lotka [11] and Volterra [25] is considered to be one of the basic models between different species in nature
It is well known that delays, which occur in the interaction between predator and prey, play a complicated role on a predator–prey system
The system may be stable or unstable (Hopf bifurcation occurs) with different delay τ, which means that the delay τ has great effect on the dynamics of the system, and it may affect the survival of the populations
Summary
The predator–prey model first proposed by Lotka [11] and Volterra [25] is considered to be one of the basic models between different species in nature. Ives and Dobson [7] proposed a predator–prey model to describe anti-predator behaviour. It is well known that delays, which occur in the interaction between predator and prey, play a complicated role on a predator–prey system It can cause the loss of stability and can induce various oscillations, periodic solutions [4, 5, 29, 30, 32]. In real life, the species is spatially heterogeneous, and individuals will tend to migrate towards regions of lower population density to add the possibility of survival [28] For this reason, diffusion cannot be ignored in studying the predator–prey system. Assume that the predator needs a gestation period τ to give birth In this case, the corresponding model with homogeneous Neumann boundary conditions is as follows:.
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