Contraction Dynamics in Heterogeneous Spatial Environments

TL;DR

This paper investigates how spatial heterogeneity influences the long-term behavior of reaction-diffusion systems, using contraction theory to establish conditions for exponential convergence and applying findings to biochemical models.

Contribution

It introduces a novel analysis of reaction-diffusion systems with spatially varying reactions and diffusion using contraction theory, providing explicit conditions for convergence.

Findings

Spatial heterogeneity affects asymptotic behavior of RD systems.

Contraction conditions guarantee exponential convergence regardless of initial conditions.

Application to biochemical systems shows how heterogeneity modulates effective reaction rates.

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.