Hopf bifurcation in a memory-based diffusion competition model with spatial heterogeneity

TL;DR

The paper investigates a diffusion-based Lotka-Volterra competition model incorporating self-memory effects and spatial heterogeneity under Dirichlet boundary conditions. By analyzing the linearized delayed system and employing a Lyapunov-Schmidt/implicit-function approach, it establishes the existence of a spatially nonconstant positive steady state and derives its stability properties. It demonstrates that Hopf bifurcation occurs as memory delay $\tau$ increases, with the stability outcome depending on memory-diffusion coefficients $(d_1,d_2)$ and environmental heterogeneity; under certain regions, the steady state remains stable for all $\tau\ge0$, while in others, $\tau$-dependent oscillations emerge. Numerical simulations corroborate the theory, revealing spatially heterogeneous steady states and Hopf-induced time-periodic patterns across different competition regimes, thereby illustrating how memory, diffusion, and heterogeneity jointly shape ecological coexistence.

Abstract

In this paper, we investigate a Lotka-Volterra competition-diffusion system with self-memory effects and spatial heterogeneity under Dirichlet boundary conditions. We focus on how memory strength influences the coexistence and stability of competing species. By analyzing the characteristic equation, we establish the existence and stability of a spatially nonhomogeneous positive steady state and demonstrate the occurrence of Hopf bifurcation as memory delay increases. Our results reveal that both weak and some opposing memory effects of two competing species promote stable coexistence, while strong memory may destabilize the system and lead to periodic oscillations. Spatial heterogeneity further enriches the dynamical behaviors. Numerical simulations are presented to confirm the theoretical results.

Figures (7)

• Figure 1: The regions of $(d_1, d_2)$ in $D_1$, $D_2$ and $D_3$, where $l_1: d_2=-\frac{\kappa_1}{\kappa_2}d_1+\frac{K_1+K_2}{\kappa_2}$, $l_2: d_2=-\frac{\kappa_1}{\kappa_2}d_1-\frac{K_1+K_2}{\kappa_2}$, $l_3: d_2=\frac{K_3\kappa_1}{K_4\kappa_2}d_1+\frac{|K_1K_3-K_2K_4|}{K_4\kappa_2}$, $l_4: d_2=\frac{K_3\kappa_1}{K_4\kappa_2}d_1-\frac{|K_1K_3-K_2K_4|}{K_4\kappa_2}$.
• Figure 2: Left: the resource distributed function $r_1(x)=\cos{x}+1$, and the principle eigenvalue function $\phi_*(x)$ of \ref{['2.3']}; Right: the resource distributed function $r_2(x)=\sin{x}+1$, and the principle eigenvalue function $\psi_*(x)$ of \ref{['2.4']}
• Figure 3: The regions of $(d_1, d_2)$ satisfying $\mathbf{(H_5)}$ based on Figure \ref{['f1.1']}, where $l_5: d_2=\frac{K_2\kappa_1}{K_1\kappa_2}d_1$, $l_6: d_2=\frac{(K_2-K_3)\kappa_1}{(K_1-K_4)\kappa_2}d_1$.
• Figure 4: (a) Numerical simulations of \ref{['1']} for $P_1\in D_2\cap (\mathbb R^+\times\mathbb R^-)$, the positive steady state $(u_s, v_s)$ is locally asymptotically stable for $\tau\ge 0$; (b) Numerical simulations of \ref{['2.6']} for $P_2\in D_2\cap (\mathbb R^+\times\mathbb R^+)$, the positive steady state $(u_s, v_s)$ is locally asymptotically stable for $\tau\ge 0$.
• Figure 5: Numerical simulations of \ref{['2.6']} for $P_3\in D_3$ satisfying $\mathbf{(H_5)}$, (a) $\tau=4<\tau_0=4.6458$, the positive steady state $(u_s, v_s)$ is locally asymptotically stable; (b) $\tau=10>\tau_0=4.6458$, the positive steady state $(u_s, v_s)$ converges to a spatially non-homogeneous time-periodic solution.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof