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

Abstract

We derive a simple sufficient condition for the local asymptotic stability of spatially discrete, continuous-time reaction-diffusion systems of networked dynamical systems at a homogeneous equilibrium point. The framework explicitly accommodates \emph{heterogeneous} local dynamics -- patches at different nodes governed by structurally distinct functional forms -- a setting not covered by the classical bookkeeping reduction of Jansen and Lloyd (2000), which requires identical patch dynamics, nor by the Master Stability Function of Pecora and Carroll (1998), which is restricted to identical nodes. The stability condition separates cleanly into two independent components: (i) a diagonal dominance criterion on the \emph{spatially averaged Jacobian} of the local patch dynamics, verifiable directly from model parameters without computing eigenvalues of the full composite system; and (ii) a lower bound on the algebraic connectivity (Fiedler value) of the network Laplacian, capturing the role of network topology. The resulting sufficient condition holds for purely conservative dispersal (standard graph Laplacians with zero row sums) and does not require any dispersal loss or mortality during transit -- a restrictive assumption appearing in the author's prior work (2021) and many classical multi-patch analyses. The theory is illustrated through metapopulation networks of predator-prey systems with heterogeneous functional responses, including a striking example in which individually unstable patches are stabilized entirely by dispersal connections.