Connecting metapopulation heterogeneity to aggregated lifetime statistics

Aggregated metapopulation lifetime statistics has been used to access stylized facts that might help identify the underlying patch-level dynamics. For instance, the emergence of scaling laws in the aggregated probability distribution of patch lifetimes can be associated to critical phenomena, in which the correlation length among system units tends to diverge. Nevertheless, an aggregated approach is biased by patch-level variability, a fact that can blur the interpretation of the data. Here, I propose a weakly-coupled metapopulation model to show how patch variability can solely trigger qualitatively different lifetime probability distribution at the aggregated level. In a generalized approach, I obtain a two-way connection between the variability of a certain patch property (e.g. carrying capacity, environment condition or connectivity) and the aggregated lifetime probability distribution. Furthermore, for a particular case, assuming that scaling laws are observed at the aggregated-level, I speculate the heterogeneity that could be behind it, relating the qualitative features the variability (mean, variance and concentration) to the scaling exponents. In this perspective, the application points to the possibility of equivalence between heterogeneous weakly-coupled metapopulations and homogeneous ones that exhibit critical behavior.