Mean exit time and escape probability for the stochastic logistic growth model with multiplicative α-stable Lévy noise

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker–Planck equation for fish population [Formula: see text]. In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker–Planck equation as growth rate [Formula: see text], carrying capacity [Formula: see text], intensity of Gaussian noise [Formula: see text], noise intensity [Formula: see text] and stability index [Formula: see text] vary. The MET from the interval [Formula: see text] at the right boundary is finite if [Formula: see text]. For [Formula: see text], the MET from [Formula: see text] at this boundary is infinite. A larger stability index [Formula: see text] is less likely leading to the extinction of the fish population.

Stochastic population dynamics driven by mutant interactors

Spontaneous random mutations are an important source of variation in populations. Many evolutionary models consider mutants with a fixed fitness chosen from a certain fitness distribution without considering any interactions among the residents and mutants. Here, we go beyond this and consider “mutant interactors”, which lead to new interactions between the residents and invading mutants that can affect the carrying capacity and the extinction risk of populations. We model microscopic interactions between individuals by using a dynamical payoff matrix and analyze the stochastic dynamics of such populations. New interactions drawn from invading mutants can drive the population away from the previous equilibrium, and lead to changes in the population size — the population size is an evolving property rather than a fixed number or externally controlled variable. We present analytical results for the average population size over time and quantify the extinction risk of the population by the mean time to extinction.

Evaluating an icon of population persistence: the Devil's Hole pupfish

The Devil's Hole pupfish Cyprinodon diabolis has iconic status among conservation biologists because it is one of the World's most vulnerable species. Furthermore, C. diabolis is the most widely cited example of a persistent, small, isolated vertebrate population; a chronic exception to the rule that small populations do not persist long in isolation. It is widely asserted that this species has persisted in small numbers (less than 400 adults) for 10 000–20 000 years, but this assertion has never been evaluated. Here, we analyse the time series of count data for this species, and we estimate time to coalescence from microsatellite data to evaluate this hypothesis. We conclude that mean time to extinction is approximately 360–2900 years (median 410–1800), with less than a 2.1% probability of persisting 10 000 years. Median times to coalescence varied from 217 to 2530 years, but all five approximations had wide credible intervals. Our analyses suggest that Devil's Hole pupfish colonized this pool well after the Pleistocene Lakes receded, probably within the last few hundred to few thousand years; this could have occurred through human intervention.

Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

The quasistationary distribution of the stochastic logistic model

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.