The Critical Patch Size Problem in Random Graphs

TL;DR

This paper investigates the critical patch size for population survival in random graph habitats, establishing a spectral threshold that determines viability with high probability.

Contribution

It introduces the first spectral theory for critical patch size on graphs, linking Dirichlet eigenvalues to habitat viability in random graph models.

Findings

Law of large numbers for Dirichlet eigenvalues in random habitats

Emergence of a sharp threshold for viability based on reaction-diffusion ratio

High probability of either survival or extinction depending on the threshold

Abstract

The problem of {\it critical patch size} -- a threshold condition for population persistence -- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.