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This paper develops a theoretical framework to model autocatalytic processes where replication occurs at interfaces, considering particle movement in a bulk medium and interaction with surfaces where they can disappear or reproduce. The authors derive a renewal-type nonlinear integral equation for the generating function of the population size and an equivalent Fokker-Planck equation with nonlinear Robin-type boundary conditions to describe the dynamics. The analysis reveals distinct dynamical regimes and universal scaling laws governing population extinction or explosive growth based on the interplay between loss and replication at surfaces.
Surface-mediated autocatalysis isn't just about growth; this model reveals how surface interactions can tip the balance between explosive population growth and complete extinction.
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.
