Reaction-diffusion kinetics in growing domains

TL;DR

This paper reviews recent advances in reaction-diffusion models on growing domains, focusing on diffusion processes and simple annihilation reactions, with implications for biological systems like development and pattern formation.

Contribution

It introduces new insights into reaction-diffusion kinetics on dynamically growing domains, extending classical models to more realistic biological scenarios.

Findings

Analysis of diffusion in growing domains

Study of encounter-controlled annihilation reactions

Framework for complex reaction-diffusion systems

Abstract

Reaction-diffusion models have been used over decades to study biological systems. In this context, evolution equations for probability distribution functions and the associated stochastic differential equations have nowadays become indispensable tools. In population dynamics, say, such approaches are utilized to study many systems, e.g., colonies of microorganisms or ecological systems. While the majority of studies focus on the case of a static domain, the time-dependent case is also important, as it allows one to deal with situations where the domain growth takes place over time scales that are relevant for the computation of reaction rates and of the associated reactant distributions. Such situations are indeed frequently encountered in the field of developmental biology, notably in connection with pattern formation, embryo growth or morphogen gradient formation. In this chapter, we review some recent advances in the study of pure diffusion processes in growing domains. These results are subsequently taken as a starting point to study the kinetics of a simple reaction-diffusion process, i.e., the encounter-controlled annihilation reaction. The outcome of the present work is expected to pave the way for the study of more complex reaction-diffusion systems of possible relevance in various fields of research.