[Paper Review] Growth Models Under Uniform Catastrophes

Joan Amaya, Valdivino V. Junior+2 more|arXiv (Cornell University)|Feb 6, 2026

Mathematical and Theoretical Epidemiology and Ecology Models0 citations

TL;DR

The paper analyzes stochastic growth models of colony-structured populations under uniform catastrophes, deriving survival probabilities and mean extinction times, with and without dispersion, and compares uniform with binomial and geometric catastrophes.

ABSTRACT

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.

Motivation & Objective

• Motivate the study of stochastic population growth under catastrophic events in colonial structures.
• Develop and analyze growth models with uniform catastrophes, including dispersion/no-dispersion scenarios and spatial constraints.
• Derive explicit survival probabilities and mean times to extinction for the proposed models.
• Compare uniform catastrophes with binomial and geometric catastrophes to quantify their impact on persistence.

Proposed method

• Model population growth as a Poisson process with rate λ for colonies; catastrophes occur as a Poisson process with rate 1.
• Define three growth models: no dispersion (single colony persists post-catastrophe), dispersion with spatial restriction, and dispersion with no spatial restrictions.
• Compute the distribution of survivors N after a uniform catastrophe; P(N=n) and E(s^N) are given, with E(N)=λ/2.
• Establish extinction criteria and mean extinction times using Markov chain/branching process methods, including Foster’s theorem for non-dispersion case.
• Derive explicit extinction probabilities for d=2 and d=3 under dispersion with spatial restriction; provide formulas for ψ_d and E[τ_d].
• Compare uniform catastrophes to geometric and binomial catastrophes in dispersion-no-spatial model, via extinction probabilities.

Figure 1. Comparison between extinction probabilities in models with uniform catastrophes and geometric catastrophes.

Experimental results

Research questions

• RQ1 Under uniform catastrophes, what is the survival probability versus time to extinction for growth models with and without dispersion?
• RQ2 How do spatial constraints and network structure (trees with degree d) influence the survival-extinction transition under dispersion?
• RQ3 What are the exact extinction probabilities and mean extinction times for small-d (e.g., d=2, d=3) trees under dispersion?
• RQ4 How does dispersion compare to binomial and geometric catastrophes in terms of population persistence under dispersion/no spatial constraints?
• RQ5 What is the limiting behavior of the critical growth rate λ_d as the environment becomes high-dimensional or unconstrained?

Key findings

• Dispersion dramatically affects persistence: without dispersion the population goes extinct almost surely for all λ>0, with finite mean extinction time.
• For dispersion on homogeneous trees, survival occurs iff (d^2/(d-1)) ln((λ+d)/d) < λ, yielding a phase transition in λ and d.
• For d=2, survival iff 4 ln(1+λ/2) < λ; extinction probability ψ_2 is given explicitly and E[τ_2] is derived in certain regimes.
• For d=3, survival iff (9/2) ln(1+λ/3) < λ with explicit ψ_3 and E[τ_3] formulas.
• Dispersion without spatial restrictions yields survival iff λ>2, with ψ_* given by solving ln[1+λ(1-s)] = λs(1-s) and E[τ_*] given by an integral expression.
• Critical parameters λ_d decrease with d and converge to 2 as d→∞, indicating high-dimensional environments approximate unrestricted dispersion.
• Compared to geometric catastrophes, uniform catastrophes are more severe in terms of extinction probability; compared to binomial catastrophes, uniform catastrophes are more severe except in certain parameter regimes (p<1/3).

Figure 2. Comparison between extinction probabilities in models with uniform catastrophes and binomial catastrophes.