Contraction Dynamics in Heterogeneous Spatial Environments

Abstract

Understanding the asymptotic behavior of reaction-diffusion (RD) systems is crucial for modeling processes ranging from species coexistence in ecology to biochemical interactions within cells. In this work, we analyze RD systems in which diffusion is modeled using the $\theta$-diffusion framework, while the reaction dynamics are spatially varying. We demonstrate that spatial heterogeneity affects the asymptotic behavior of such systems. Using contraction theory, we derive conditions that guarantee the exponential convergence of system trajectories, regardless of initial conditions. These conditions explicitly account for the influence of spatial heterogeneity in both the diffusion and reaction terms. As an application, we study a biochemical system and derive the quasi-steady-state (QSS) approximation, illustrating how spatial heterogeneity modulates the effective binding rates of biomolecular species.