Growth Models Under Uniform Catastrophes

We consider stochastic growth models for populations organized in colonies and subject to uniform catastrophes. To assess population viability, we analyze scenarios in which individuals adopt dispersion strategies after catastrophic events. For these models, we derive explicit expressions for the survival probability and the mean time to extinction, both with and without spatial constraints. In addition, we complement this analysis by comparing uniform catastrophes with binomial and geometric catastrophes in models with dispersion and no spatial restrictions. Here, the terms uniform, binomial and geometric refer to the probability distributions governing the number of individuals that survive immediately after a catastrophe. This comparison allows us to quantify the impact of different types of catastrophic events on population persistence.

💡 Research Summary

This paper investigates stochastic growth models for populations organized in colonies that are subject to uniform catastrophes, focusing on how post‑catastrophe dispersion strategies affect survival probability and mean time to extinction. Three main families of models are introduced.

The first model (no dispersion) treats a single colony as a continuous‑time Markov process C(λ) with birth rate λ and catastrophe rate 1. When a catastrophe occurs, the colony size i is reduced uniformly to any j∈{0,…,i‑1} with probability 1/i. Using the infinitesimal generator, Theorem 2.2 proves that for any λ>0 the process goes extinct almost surely; without dispersion the population cannot recover from uniform catastrophes.

The second and third models incorporate dispersion on an infinite rooted d‑regular tree T⁺_d (degree d+1 except at the root). After each catastrophe, each surviving individual independently selects one of the d neighboring vertices farthest from the root to attempt founding a new colony; if several individuals choose the same vertex, only one succeeds and the rest die. This yields a branching‑like process C_d(λ). Coupling arguments show that the extinction probability ψ_d is non‑increasing in both d and λ. Theorem 2.3 gives a precise survival condition:

  (d·2/(d‑1))·ln