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On the Stability of Discrete Reaction-Diffusion System of Networked Dynamical Systems

Dinesh Kumar

TL;DR

The paper addresses the stability of discrete reaction-diffusion networks of $m$ coupled dynamical systems in $n$ variables around a homogeneous equilibrium, formalized as $\dot{x}=f(x)-Lx$ with $L=\mathcal{L}_1\oplus\dots\oplus\mathcal{L}_n$. The approach linearizes around the equilibrium to $\dot{x}=(Df(\bar{x})-L)x$, then uses a block-diagonalization via $x=Py$ to obtain $\dot{y}=(P^{-1}Df(\bar{x})P-\Lambda)y$ and applies Gershgorin disc arguments together with Weyl's monotonicity. The main result provides a sufficient condition: the averaged local Jacobians must be diagonally dominant with negative diagonal entries, and the network connectivity must satisfy $\lambda_2\ge\tau$. This is illustrated with ecological metapopulation networks, showing that stronger connectivity can stabilize homogeneous equilibria even when local dynamics are unstable, offering a design principle for robust stability in spatially structured systems.

Abstract

Paper provides a sufficient condition for the local stability of the reaction-diffusion system of networked dynamical systems at its homogeneous equilibrium point. It is illustrated through an ecological meta-populations (spatial-structured populations) network of predator-prey systems.

On the Stability of Discrete Reaction-Diffusion System of Networked Dynamical Systems

TL;DR

The paper addresses the stability of discrete reaction-diffusion networks of coupled dynamical systems in variables around a homogeneous equilibrium, formalized as with . The approach linearizes around the equilibrium to , then uses a block-diagonalization via to obtain and applies Gershgorin disc arguments together with Weyl's monotonicity. The main result provides a sufficient condition: the averaged local Jacobians must be diagonally dominant with negative diagonal entries, and the network connectivity must satisfy . This is illustrated with ecological metapopulation networks, showing that stronger connectivity can stabilize homogeneous equilibria even when local dynamics are unstable, offering a design principle for robust stability in spatially structured systems.

Abstract

Paper provides a sufficient condition for the local stability of the reaction-diffusion system of networked dynamical systems at its homogeneous equilibrium point. It is illustrated through an ecological meta-populations (spatial-structured populations) network of predator-prey systems.

Paper Structure

This paper contains 4 sections, 2 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model Stability
  3. Examples
  4. Conclusion

Key Result

Theorem 2.1

Let $f(x)$ be a continuously differentiable function and the Jacobian of each dynamics $f_{i}~(i=1,\cdots,n)$ (evaluated at the equilibrium point) has the diagonal dominance property with negative diagonal entries. Then, the dynamics of coupled dynamical systems in the network is locally stable if n

Figures (2)

  • Figure 1: A schematic representation of the 3-patch spatial network of predator-prey systems.
  • Figure 2: Figure shows the 5-patch spatial predator-prey system with the dispersal connections. Dispersal rates between patches denoted by $d^1_{ij}(=d^1_{ji})$ and $d^2_{ij}(=d^2_{ji})$, are of prey and predator species respectively.

Theorems & Definitions (4)

  • Theorem 2.1
  • proof
  • Corollary 2.1.1
  • proof