The critical patch size problem on networks

The notion of critical patch size in the context of population dynamics on strongly heterogeneous media, namely networks, is investigated. Individuals reproduce at the nodes via logistic growth and disperse across the network according to the graph Laplacian operator. A subset of nodes is designated as sinks, representing zones where the population cannot survive. In contrast to homogeneous or mildly heterogeneous continuous domains, where analytical expressions for the critical patch size can often be derived, the high heterogeneity and discrete nature of graphs pose significant analytical challenges. Starting from simple cases such as discrete lattices and lattices with multiple links, where exact or approximate critical dimensions can be computed, we extend the analysis to random networks. In this setting, we derive bounds on spectral graph parameters that guarantee the leading eigenvalue of the linearized evolution operator remains positive, a necessary condition for population persistence. Our findings, representing to our knowledge the first general study on critical patch size in graphs, highlight how network topology and sink distribution govern survival thresholds, with potential applications to ecological networks, synthetic biology, and the modeling of neurodegenerative disease propagation on brain connectomes.