Finite-size scaling of survival statistics in metapopulation models

Spatial metapopulation models are fundamental to theoretical ecology, enabling us to study how landscape structure influences global species dynamics. Traditional models, including recent generalizations, often rely on the deterministic limit of stochastic processes, assuming large population sizes. However, stochasticity, arising from dispersal events and population fluctuations, profoundly shapes ecological dynamics. In this work, we extend the classical metapopulation framework to account for finite populations, examining the impact of stochasticity on species persistence and dynamics. Specifically, we analyze how the limited capacity of local habitats influences survival, deriving analytical expressions for the finite-size scaling of the survival probability near the critical transition between survival and extinction. Crucially, we demonstrate that the deterministic metapopulation capacity plays a fundamental role in the statistics of survival probability and extinction time moments. These results provide a robust foundation for integrating demographic stochasticity into classical metapopulation models and their extensions.