Global stability and stationary pattern of a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response

Abstract

This paper is concerned with a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response subject to the homogeneous Neumann boundary condition. Firstly, by upper and lower solutions method, we prove the global asymptotic stability of the unique positive constant steady state solution. Secondly, introducing the cross diffusion, we obtain the existence of non-constant positive solutions. The results demonstrate that under certain conditions, even though the unique positive constant steady state is globally asymptotically stable for the model with self-diffusion, the non-constant positive steady states can exist due to the emergency of cross-diffusion, that is to say, cross-diffusion can create stationary pattern. Finally, using the bifurcation theory and treating cross diffusion as a bifurcation parameter, we obtain the existence of positive non-constant solutions.

Traveling waves of a modified Holling-Tanner predator-prey model with degenerate diffusive

“…It is worth mentioning that the classical ways of constructing the upper-lower solutions (see [14,16,17,18,19] and references therein) are not feasible for model (1.3) due to the degenerate diffusion. We replace the function construction method by solving differential equations to locate a pair of upper-lower solutions for model (1.3).…”

“…indicates that predator tend to catch other prey when the number of their preferred prey is insufficient [12]. More explanations of these parameters can be found, among others, in [13,14,15]. The term (v n v x ) x with n > 0 represent degenerate diffusion.…”

“…If n = 0, model (1.1) with random diffusion, traveling waves, bifurcation and stability have been analyzed, see [14,16,17,18,19] and so on. Mathematically, model (1.1) may be degenerate with n > 0 and its analysis becomes delicate.…”

On the Local Stability of a Certain Class of Polynomial Differential System

“…An example of such species interaction models is the prey-predator system with competition and/or cooperation that has been modelled and studied using two dimensional system of polynomial differential equations [1,7], in electrical engineering, an electrical circuit consists of elements each of which has current, I(t) flowing through and the voltage difference, V (t). The relationship between I(t) and V (t) is modelled by equations resulting from Kirchoff's law that can be reduced to the form of Liénard's equations that take the form of ( 1), [15,9]. One can also reduce certain partial differential equations that describe control problems to a system of ordinary differential equations and study the resulting system for example in [8] and [13].…”

Dynamics of a diffusive two predators- one prey system

“…Quo and Guo [10] analyzed a diffusive and advective LG model. Li et al [31] discussed global asymptotic stability and stationary pattern of a diffusive prey-predator system with LG term and diffusion. Predator-prey models with more than two species create a major interest to the researchers.…”