Does genetic diversity help survival?

TL;DR

This paper presents a population evolution model where genetic diversity, represented by variable death probabilities, determines survival, showing that infinite expected reciprocal of death probabilities leads to unbounded population growth.

Contribution

It introduces a novel stochastic model linking genetic diversity to population survival, contrasting with traditional models with constant or uniform environmental factors.

Findings

Population size diverges if E(1/C) is infinite.

Genetic diversity enhances survival prospects.

Limit point process converges to a non-homogeneous Poisson process for certain distributions.

Abstract

We introduce the following model for the evolution of a population. At every discrete time $j\geq 0$ exactly one individual is introduced in the population and is assigned a death probability $c_j$ sampled from $C$, a fixed probability distribution. We think of $c_j$ as a genetic marker of this individual. At every time $n\geq 1$ every individual in the population dies or not independently of each other with its corresponding death probability $c_j$. We show that the population size goes to infinity if and only if $E(1/C)=\infty$. This is in sharp contrast with the model with constant $c$ and with the model in random environment (same random $c_n$ for all individuals at time $n$). Both of these models are always positive recurrent. Thus, genetic diversity does seem to help survival! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the $c'$s. For certain $C$ distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.