How fast extinction occurs in bisexual populations with size-depending mating dynamics?

TL;DR

The paper analyzes extinction timing in bisexual populations modeled by a generalized bisexual Galton-Watson process with size-dependent mating dynamics. It uses Extreme Value Theory, specifically the Peaks Over Threshold method, to approximate the tail of the time-to-extinction distribution, proving that conditioned on extinction the extinction time converges to a Gumbel-domain distribution in the discrete setting. The authors establish conditions (notably theta>ln(2)) under which the tail can be approximated by an exponential/GPD model, and they illustrate this with numerical experiments across several mating functions and offspring distributions. This approach provides a simple, data-driven tool for estimating extinction timing in populations with sexual reproduction and dynamic mating rules, with potential applications in ecology and population genetics.

Abstract

Given that extinction in a bisexual population is certain, we study a way to approximate the time when this extinction occurs. Our study is based on standard tools from Extreme Value Theory, which in practice are very easy to implement. We present the theoretical results derived from our study and provide a few numerical examples of such results.

Figures (12)

• Figure 1: Simulated values of $\tau$ for case A (a) and case B (b). The fact that not all data lays in the same rectangular region suggests that the right endpoint of this distribution is not finite, which coincides with Proposition \ref{['omegainfinito']}. Furthermore, there are very few values that drastically differ from the others, which suggests that this data set comes from a light-tailed distribution (in agreement with Proposition \ref{['extincionligera']}). The data corresponding to (a) has an estimated mean of 32.29416 with an estimated standard deviation of 7.791591, while data for (b) have estimated mean and standard deviation given respectively by 41.46776 and 5.274994.
• Figure 2: Goodness of fit of a GPD for case A (a) and case B (b). The Q-Q plots show that the behavior of the data indeed looks like a discretization of a data set to which the GPD would fit well.
• Figure 3: Goodness of fit of an exponential distribution to the data under case A (a) and case B (b). Again we see the ladder-like behavior of the data, which indicates a nice fitting of this distribution to the given data set.
• Figure 4: Simulated values of $\tau$ under case A (a) and case B (b) The data corresponding to (a) has an estimated mean of 4.39602 with an estimated standard deviation of 1.883328, while data for (b) have estimated mean and standard deviation given respectively by 4.29541 and 1.684622. In both cases, the dispersion of the data seems to support the fact that the distribution behind each data set has an infinite endpoint and a light-tail.
• Figure 5: Goodness of fit of a GPD for case A (a) and case B (b). The Q-Q plots support the fac that data come from a discretizatión of a distribution in some maximal domain of attraction.

Theorems & Definitions (14)

• Definition 1
• Proposition 1
• Proposition 2
• Lemma 1
• Remark 1
• Lemma 2
• Theorem 1
• Theorem 2
• Remark 2
• Lemma 3