Robustness Against Extinction by Stochastic Sex Determination in Small Populations

Sexually reproducing populations with small number of individuals may go extinct by stochastic fluctuations in sex determination, causing all their members to become male or female in a generation. In this work we calculate the time to extinction of isolated populations with fixed number $N$ of individuals that are updated according to the Moran birth and death process. At each time step, one individual is randomly selected and replaced by its offspring resulting from mating with another individual of opposite sex; the offspring can be male or female with equal probability. A set of $N$ time steps is called a generation, the average time it takes for the entire population to be replaced. The number k of females fluctuates in time, similarly to a random walk, and extinction, which is the only asymptotic possibility, occurs when k=0 or k=N. We show that it takes only one generation for an arbitrary initial distribution of males and females to approach the binomial distribution. This distribution, however, is unstable and the population eventually goes extinct in 2^N/N generations. We also discuss the robustness of these results against bias in the determination of the sex of the offspring, a characteristic promoted by infection by the bacteria Wolbachia in some arthropod species or by temperature in reptiles.

💡 Research Summary

The paper investigates a subtle but potentially fatal source of extinction in small sexually reproducing populations: stochastic fluctuations in the sex of offspring that can, by chance, drive an entire generation to consist of only males or only females. The authors model an isolated population of fixed size N using the classic Moran birth‑death process. At each elementary time step one individual is chosen uniformly at random, mates with a randomly selected partner of the opposite sex, and is replaced by an offspring whose sex is male with probability p (p = ½ in the unbiased case) and female with probability 1‑p. A “generation” consists of N such elementary steps, i.e., the time required for the whole population to be replaced.

The state of the system can be described by the number k of females (0 ≤ k ≤ N). The dynamics of k form a one‑dimensional Markov chain with transition probabilities

 P(k→k+1) =