Reaction-diffusion processes with surface autocatalysis

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 diffuse through a bulk medium and interact with surface regions, where they may either disappear or reproduce through branching, cloning, or splitting. The interplay between loss and replication at surfaces gives rise to rich population dynamics. Here, we develop a general theoretical framework for diffusion-mediated autocatalytic processes at surfaces. We derive a nonlinear renewal-type integral equation for the generating function of the population size, which provides access to its full probability distribution and integer moments. We further establish an equivalent description in terms of a Fokker-Planck equation with nonlinear Robin boundary conditions that encode surface autocatalytic reactions. Our results identify universal asymptotic regimes and provide a unified framework to predict when surface activity promotes extinction or explosive growth of the population. The developed quantitative framework opens new avenues for analyzing catalytic efficiency, metabolic regulation, and population persistence in spatially heterogeneous environments.