Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

Abstract

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity

“…In order to reflect the real dynamical behaviors of models that depend on the past history of systems, it is reasonable to incorporate time delays into the systems [11,13,20,34]. Especially in mathematical biology, many models of population dynamics can be described by delayed reaction-diffusion equations [2,4,5,8,9,10,12,14,19,35,36]. Lam and Ni [18] investigated the interactions between diffusion and heterogeneity of the environment in the following classical diffusive Lotka-Volterra type model:…”

Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

“…Guo and Yan [13] investigated the following Lotka-Volterra competition system with nonlocal delay effect where By means of Lyapunov-Schmidt reduction, the normal form theory and the center manifold reduction, the existence and stability of spatially nonhomogeneous steadystate solutions and bifurcation direction of Hopf bifurcating periodic orbits are given. Then, Yan and Guo [31] further extended the above conclusion to Lotka-Volterra model with cross-diffusion and delay effect. And Han and Wang [14] also studied the following Lotka-Volterra competition diffusion system with nonlocal term where * * u and * * v are similar to model (1.1) with (1.2).…”

Bifurcation and stability of a two-species diffusive Lotka-Volterra model

“…For this purpose, define a nonlinear mapping F : Clearly, the Crandall-Rabinowitz bifurcation theorem [3,4] does not work here. Hence, we shall resort to Lyapunov-Schmidt reduction method [12,13,26,14,28,44,50,51]. Firstly, we define the operator P by P U = u, ϕ 1 Φ + v, ψ 1 Ψ for U = (u, v) T ∈ X and decompose X as X = X 1 ⊕X 2 with X 1 = P X and X 2 = (I−P )X.…”