Hopf bifurcations in a reaction-diffusion model with a general advection term and delay effect

TL;DR

This work analyzes a reaction-diffusion-advection population model with time-delayed growth on bounded domains, capturing logistic, food-limited, and weak Allee dynamics. Using Lyapunov-Schmidt reduction, spectral analysis of a non-self-adjoint operator, and center-manifold/normal-form theory, it proves the existence of spatially nonhomogeneous steady states near the principal eigenvalue and demonstrates Hopf bifurcations from these states as delay varies. The authors deriveDirection and stability criteria for the bifurcating spatially nonhomogeneous periodic orbits and provide explicit formulas for the normal-form coefficients. Numerical simulations on food-limited and weak Allee models confirm the theoretical predictions, illustrating delay-induced transitions to stable or unstable spatially structured oscillations with practical relevance to ecological systems.

Abstract

This paper investigates a class of reaction-diffusion population models defined on a bounded domain, characterized by a general time-delayed per capita growth rate and a general advection term. Notably, the growth rate encompasses both Logistic-type and weak Allee effect-type dynamical behaviors. By applying the Lyapunov method, we establish the existence of spatially inhomogeneous steady states when a parameter approaches the principal eigenvalue of a non-self-adjoint elliptic operator. A detailed analysis of the characteristic equation further confirms the existence of Hopf bifurcations originating from these steady states. Subsequently, by applying center manifold reduction and normal form theory, we ascertain the direction of these Hopf bifurcations and the stability of the resulting periodic orbits. Finally, the proposed general theoretical results are successfully applied to a "food-limited" population model and a weak Allee effect-driven population model, each of which incorporates diffusion, time delay, and advection, thus confirming the validity of our approach.

Figures (2)

• Figure 1: (a) $\tau = 8$, the solution approaches to the positive steady state. (b) $\tau = 10$, the solution still approaches to the positive steady state but with noticeable oscillations. (c) $\tau = 12$, the solution converges to a time-periodic solution with small oscillations. (d) $\tau = 20$, the solution converges to a time-periodic solution and the period and amplitude of the stable periodic solution both increase as the time delay $\tau$ increases.
• Figure 2: (a) for $\tau$ = 1.5, the solution converges to positive steady state $u_\lambda$.(b) for $\tau$ = 5.5 , the solution approaches $u_\lambda$ with oscillations. (c)for $\tau$ = 9.5, the solution converges to a time-periodic solution.(d)for $\tau$ = 15.5the solution asymptotically converges to a time-periodic solution. Both the period and amplitude of the emerging stable periodic orbit increase with the time delay $\tau$.

Theorems & Definitions (32)

• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof