[Paper Review] The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system

Yongli Song, Yahong Peng+1 more|arXiv (Cornell University)|Apr 1, 2021

Mathematical and Theoretical Epidemiology and Ecology Models28 references4 citations

TL;DR

This paper develops a novel normal form algorithm for analyzing delay-induced spatially inhomogeneous Hopf bifurcations in reaction-diffusion systems with memory-based diffusion, where the delay appears in the nonlinear diffusion term. The method enables determination of the direction and stability of mode-1 and mode-2 Hopf bifurcations, and numerical results confirm the existence of stable spatially inhomogeneous periodic solutions, including a transition from unstable mode-2 to stable mode-1 patterns.

ABSTRACT

The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf bifurcations of such systems, we develop an effective new algorithm where the memory delay is treated as the perturbation parameter. To illustrate the effectiveness of the algorithm, we consider a diffusive predator-prey system with memory-based diffusion and Holling type-II functional response. By employing this newly developed procedure, we investigate the direction and stability of the delay-induced mode-1 and mode-2 Hopf bifurcations. Numerical simulations confirm our theoretical findings, that is the existence of stable spatially inhomogeneous periodic solutions with mode-1 and mode-2 spatial patterns, and the transition from the unstable mode-2 spatially inhomogeneous periodic solution to the stable mode-1 spatially inhomogeneous periodic solution.

Motivation & Objective

• To address the lack of normal form computation methods for reaction-diffusion systems with nonlinear memory-based diffusion and delay in the diffusion term.
• To develop a systematic algorithm capable of analyzing spatially inhomogeneous Hopf bifurcations induced by memory delay.
• To apply the algorithm to a diffusive predator-prey model with Holling type-II functional response and memory-based diffusion.
• To determine the direction and stability of delay-induced mode-1 and mode-2 Hopf bifurcations.
• To numerically validate the theoretical findings on the existence and transition between spatially inhomogeneous periodic solutions.

Proposed method

• The authors generalize existing normal form algorithms to handle nonlinear diffusion terms and delay in the diffusion component, treating memory delay τ as the perturbation parameter.
• They derive the characteristic equation at the positive equilibrium and compute the critical delay values τ₁,₀ ≈ 5.1033 and τ₂,₀ ≈ 1.8398 for mode-1 and mode-2 bifurcations.
• The method involves computing the center manifold and normal form coefficients K₁ and K₂ using a systematic procedure adapted to the nonlinearity and delay in the diffusion term.
• The direction and stability of the Hopf bifurcations are determined by the signs of K₁ and K₂, with K₁ > 0 and K₂ < 0 indicating supercritical and stable bifurcations.
• Numerical simulations are performed using the full system with specific parameters to visualize spatiotemporal dynamics and validate theoretical predictions.
• The algorithm is applied to a predator-prey model with memory-based diffusion, where d₂₁ and τ are varied to explore bifurcation curves and pattern transitions.

Figure 1: Stability region and Hopf bifurcation curves in $d_{21}-\tau$ plane. The dotted region is the stability region and $\tau=\tau_{k,0},k=1,2,3$ , are Hopf bifurcation curves. Hopf bifurcation curves $\tau=\tau_{1,0}$ and $\tau=\tau_{2,0}$ intersect at the point $P(4.1354,4.0292)$ . The points

Experimental results

Research questions

• RQ1 How can the normal form be computed for Hopf bifurcations induced by memory delay in reaction-diffusion systems with nonlinear memory-based diffusion terms?
• RQ2 What determines the direction and stability of delay-induced mode-1 and mode-2 spatially inhomogeneous Hopf bifurcations in such systems?
• RQ3 Can stable spatially inhomogeneous periodic solutions with distinct spatial patterns (mode-1 and mode-2) be analytically and numerically confirmed?
• RQ4 What dynamical transitions occur between different spatial patterns as the delay τ increases beyond critical values?
• RQ5 How do interactions between mode-1 and mode-2 Hopf bifurcations lead to complex dynamics such as pattern transitions or quasi-periodic behavior?

Key findings

• The mode-1 spatially inhomogeneous Hopf bifurcation at τ₁,₀ ≈ 5.1033 is supercritical and stable, with K₁ ≈ 0.0597 > 0 and K₂ ≈ -1.5624 < 0.
• The mode-2 spatially inhomogeneous Hopf bifurcation at τ₂,₀ ≈ 1.8398 is also supercritical and stable, with K₁ ≈ 0.1733 > 0 and K₂ ≈ -2.2283 < 0.
• Numerical simulations confirm the existence of stable spatially inhomogeneous periodic solutions with mode-1 and mode-2 spatial patterns for τ > τ₁,₀ and τ > τ₂,₀, respectively.
• A transition from unstable mode-2 to stable mode-1 spatially inhomogeneous periodic solutions is observed numerically for τ ≈ 5.2 and d₂₁ ≈ 4.3.
• Transiently unstable quasi-periodic patterns are observed during the transition from mode-2 to mode-1 patterns, indicating complex dynamics near the double Hopf point.
• The interaction of mode-1 and mode-2 Hopf bifurcations leads to rich dynamics, including pattern transitions and potential emergence of two- or three-dimensional invariant tori.

Figure 4: The spatial-temporal dynamics of system ( 3.1 ) with the parameters $a=1,~{}b=\frac{3}{10},~{}c=\frac{1}{10},~{}d_{11}=\frac{6}{10},~{}d_{22}=\frac{8}{10},~{}\ell=2$ and $\left(d_{21},\tau\right)$ being the point $P_{4}(4.3,5.2)$ of Fig. 1 far from the Hopf bifurcation curves. Pattern tran