We consider the extinction regime in the spatial stochastic logistic model in $\mathbb{R}^d$ (a.k.a. Bolker--Pacala--Dieckmann--Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality---the smallest constant death rate which ensures the extinction of the population---as a function of the mean-field scaling parameter $\varepsilon>0$. We find the leading term of the asymptotic expansion (as $\varepsilon\to0$) of the critical mortality which is apparently different for the cases $d\geq3$, $d=2$, and $d=1$.
In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment conv…
We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as es…
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 X(t). 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 equa…
Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size $N$. We study the behaviour of the process as the population size $N$ tends to infinity. Our results cover the entire subcritical regime, including the "barely subcr…
We propose and study a stochastic capital-labour model with logistic growth function. First, we show that the model has a unique positive global solution. Then, using the Lyapunov analysis method, we obtain conditions for the extinction of the total labour force. Furthermore, we also prove sufficient conditions for their persistence in mean. Finally, we illustrate our theoretical results through numerical simulations.
In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to the current population size. Using the direct Lyapunov method, we established the global properties of this stochastic differential equation. In particular, we found that solutions of the equation oscillate around an interval, and explicitly found the end points of this interval. Moreover, we found that, if the magnitude of the noise exceeds a certain critical …
The stochastic SIRS model is a continuous-time Markov chain modelling the spread of infectious diseases with temporary immunity, in a homogeneously-mixing population of fixed size $N$. We study the scaling behaviour of the extinction time of stochastic SIRS models as $N$ tends to infinity. When the initial size of infected population is small, we obtain the closed-form expression of the asymptotic distribution of this extinction time, and compare it with the data from Monte Carlo simulation.
We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. The spread of each strain in the absence of the other one is described by the stochastic logistic SIS epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (tha…
In this article we use a criterion for the integrability of paths of one-dimensional diffusion processes from which we derive new insights on allelic fixation in several situations. This well known criterion involves a simple necessary and sufficient condition based on scale function and speed measure. We provide a new simple proof for this result and also obtain explicit bounds for the moments of such integrals. We also extend this criterion to non-homogeneous processes by use of Girsanov's transform. We apply our results to multi-type population dynamics: using the criterion with appropriate…
