Abstract
This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.
Highlights
Predator–prey models are basic differential equation models for describing the interactions between two species, and are of great interest to many researchers in mathematics and ecology
8 Conclusions In this paper, we mainly have dealt with a predator–prey model along with a spatial diffusion, Smith growth rate and herd behavior, subject to Neumann boundary condition
We have discussed the stability of positive constant equilibria, and the existence and stability of Hopf bifurcation near the positive constant equilibria
Summary
Predator–prey models are basic differential equation models for describing the interactions between two species, and are of great interest to many researchers in mathematics and ecology. (t), Here X(t) represents the prey density and Y (t) the predator density, parameter r is the growth rate of the prey, parameter K is its carrying capacity, parameter s denotes the death rate of the predator in the absence of prey, parameter α is the search efficiency of Y (t) for X(t), and parameter c is viewed as the biomass conversion or consumption rate This kind of model is known to us as the predator–prey model with herd behavior, and the existence of the possibility of sustained limit cycles is real; what’s more, the solution behavior near the origin shows to be more subtle and interesting. Many authors paid more attention to a partial differential system in the field of delay effects; this new kind of diffusion was taken into consideration in [3, 11,12,13].
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