Birth, Death, and Replication at Surfaces: Universal Laws of Autocatalytic Dynamics

TL;DR

This paper develops a unified theoretical framework for surface-mediated autocatalytic processes, revealing universal laws governing their population dynamics and conditions for growth or extinction.

Contribution

It introduces a nonlinear integral equation and a Fokker-Planck model to analyze surface autocatalytic dynamics, unifying diverse systems under a common theory.

Findings

Identifies distinct dynamical regimes and universal scaling laws.

Provides criteria for when surface activity leads to extinction or growth.

Offers quantitative insights into catalytic and biological processes.

Abstract

Autocatalytic processes underlie diverse systems in which replication is triggered at interfaces, including heterogeneous catalysis on solid substrates, enzyme activity at membranes, viral infections, biofilm growth, and spatially structured ecosystems. In a typical scenario, particles move in a bulk medium and interact with surface regions, where they may either disappear or reproduce through branching, splitting or fission. Here, we develop a general theoretical framework to understand such surface-mediated autocatalytic processes. We show that the interplay between loss and replication at surfaces gives rise to rich population dynamics. For this purpose, we derive a renewal-type nonlinear integral equation for the generating function of the population size, providing access to its full probability distribution and statistical moments. We further establish an equivalent description in terms of a Fokker-Planck equation with nonlinear Robin-type boundary conditions that encode surface reactions. Our results identify distinct dynamical regimes and universal scaling laws, and provide a unified framework to predict when surface activity promotes extinction or explosive growth. These findings offer quantitative insight into catalytic efficiency, metabolic regulation, and population persistence in spatially heterogeneous environments.