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Growth Models Under Uniform Catastrophes

Joan Amaya, Valdivino V. Junior, Fábio P. Machado, Alejandro Roldán-Correa

TL;DR

We address persistence of colonies facing uniform catastrophes by formulating three growth models: no dispersion, dispersion with spatial restrictions, and dispersion without spatial restrictions, with growth at rate λ and catastrophes at rate 1. The main method derives explicit survivor distributions via the Lerch transcendent, exact extinction criteria on trees, and closed-form or integral expressions for mean extinction times, plus a comparison with geometric and binomial catastrophe types. The results show that dispersion substantially enhances persistence, yielding clear survival conditions and finite extinction times, while the uniform catastrophe model becomes vanishingly unlikely to persist in the no-dispersion case; with dispersion, survival hinges on λ and the spatial structure, with thresholds converging to 2 as the environment becomes unconstrained. These findings provide rigorous thresholds for population persistence under different catastrophe regimes and quantify the benefits of spreading to new colonies as a resilience strategy.

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.

Growth Models Under Uniform Catastrophes

TL;DR

We address persistence of colonies facing uniform catastrophes by formulating three growth models: no dispersion, dispersion with spatial restrictions, and dispersion without spatial restrictions, with growth at rate λ and catastrophes at rate 1. The main method derives explicit survivor distributions via the Lerch transcendent, exact extinction criteria on trees, and closed-form or integral expressions for mean extinction times, plus a comparison with geometric and binomial catastrophe types. The results show that dispersion substantially enhances persistence, yielding clear survival conditions and finite extinction times, while the uniform catastrophe model becomes vanishingly unlikely to persist in the no-dispersion case; with dispersion, survival hinges on λ and the spatial structure, with thresholds converging to 2 as the environment becomes unconstrained. These findings provide rigorous thresholds for population persistence under different catastrophe regimes and quantify the benefits of spreading to new colonies as a resilience strategy.

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.
Paper Structure (13 sections, 8 theorems, 54 equations, 2 figures, 1 table)

This paper contains 13 sections, 8 theorems, 54 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Models and Results
  3. Uniform Catastrophes
  4. Growth model without dispersion
  5. Growth models with dispersion and spatial restriction
  6. Growth model with dispersion but no spatial restrictions.
  7. Discussion
  8. Dispersion as a survival strategy.
  9. A comparative analysis of catastrophe types.
  10. Proofs
  11. Growth model without dispersion
  12. Growth models with dispersion and spatial restriction
  13. Growth models with dispersion but no spatial restriction

Key Result

Proposition 2.1

Let $N$ be the number of individuals that survive an uniform catastrophe. where $\Phi$ is the Lerch Transcendent function, defined by

Figures (2)

  • Figure 1: Comparison between extinction probabilities in models with uniform catastrophes and geometric catastrophes.
  • Figure 2: Comparison between extinction probabilities in models with uniform catastrophes and binomial catastrophes.

Theorems & Definitions (16)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof : Proof of Proposition \ref{['distribuiçao_individuos']}
  • Theorem 4.1: Foster's theorem
  • ...and 6 more