Bifurcation in a Discrete Competition System

Abstract

A new difference system is induced from a differential competition system by different discrete methods. We give theoretical analysis for local bifurcation of the fixed points and derive the conditions under which the local bifurcations such as flip occur at the fixed points. Furthermore, one- and two-dimensional diffusion systems are given when diffusion terms are added. We provide the Turing instability conditions by linearization method and inner product technique for the diffusion system with periodic boundary conditions. A series of numerical simulations are performed that not only verify the theoretical analysis, but also display some interesting dynamics.

Feedback Control and Parameter Invasion for a Discrete Competitive Lotka–Volterra System

“…It is also a fact that the motion of individuals is random and isotropic, i.e., without any preferred direction, and the individuals are also absolute ones in microscopic sense, and each isolated individual exchanges materials and information by diffusion with its neighbors [3][4][5]. Then it is reasonable to consider a 2D spatially discrete reaction-diffusion system as follows [6]: …”

Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

“…Existence and nonexistence of travelling waves in those models have been recently studied in Chow [4], Chow et al [5], and Zinner [6] mostly with the cubic (or bistable, double-well) nonlinearities of the form ( ) = ( − )(1 − ), with > 0 and ∈ (0, 1) (this special case of FKPP equation is being referred to as Nagumo equation). In contrast, various reaction functions have been proposed in models without spatial interaction, for example, Xu et al [7]. Motivated by these facts, we allow for a general form of the reaction function in this paper (i.e., we do not restrict ourselves to cubic nonlinearities).…”