Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

Abstract

Communicated by M. LachowiczIn this paper, a diffusive predator-prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion-driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion-driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states.

Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping

“…In this case, the number of the captured prey by one predator will be proportional to the number of the prey population on the boundary of the prey herd, which means that is proportional to the square root of the prey population density; it is the main reason of proposing such as functional response. This functional response has been used widely, we cite for instance the papers [6,27,29,30,32,35,36]. It follows by a several of functional responses which models this interaction by including many factors such as herd shape, predator average handling time of the prey on the bounders, we take as example the papers [3,7,9,10,19,24,26,33,38].…”

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator‐Prey System

“…System (1) has been studied extensively in the past few years (see [16][17][18][19][20][21][22][23][24][25]). For example, in [16], by using fl as the bifurcation parameter, the authors showed that, as the parameter changes, the stability of the constant positive steady state varies, which induces the occurrence of Hopf bifurcating periodic solutions; however, their works tend to be numerical, rather than analytical.…”

“…By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the time delay is used to be the bifurcation parameter. In [25], in the special case when = 0, Xu and Song introduced the time delay into the system. By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the parameter is chosen as the bifurcation parameter.…”

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

“…For the diffusive Leslie–Gower predator–prey system with Dirichlet boundary condition, Zhou and Shi investigated the existence of the positive steady‐state solutions. We would also like to mention that the study on the dynamics of Lotka–Volterra predator–prey system with diffusion has also recently recovered some interest .…”