POPULATION DYNAMICS UNDER DEMOGRAPHIC AND ENVIRONMENTAL STOCHASTICITY

• 成果类型: Article
• 署名作者: Hening, Alexandru; Qi, Weiwei; Shen, Zhongwei; Yi, Yingfei
• 署名单位: Texas A&M University System; Texas A&M University Kingsville; University of Alberta
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2101
• 发表日期: 2024
• 页码: 5615-5663
• 关键词: quasi-stationary distributions; one-dimensional diffusions; random perturbations; invariant density; Markov Process; asymptotics; BOUNDARY; BEHAVIOR

摘要

The present paper is devoted to the study of the long term dynamics of diffusion processes modelling a single species that experiences both demographic and environmental stochasticity. In our setting, the long term dynamics of the diffusion process in the absence of demographic stochasticity is determined by the sign of Lambda(0), the external Lyapunov exponent, as follows: Lambda(0 )< 0 implies (asymptotic) extinction and Lambda(0 )> 0 implies convergence to a unique positive stationary distribution mu(0). If the system is of size 1/& varepsilon;(2) for small & varepsilon; > 0 (the intensity of demographic stochasticity), demographic effects will make the extinction time finite almost surely. This suggests that to understand the dynamics one should analyze the quasi-stationary distribution (QSD) mu(& varepsilon;) of the system. The existence and uniqueness of the QSD is well known under mild assumptions. We look at what happens when the population size is sent to infinity, that is, when & varepsilon;-> 0 & varepsilon;-> 0. We show that the external Lyapunov exponent still plays a key role: (1) If Lambda(0 )< 0, then mu(& varepsilon; )-> delta(0 ), the mean extinction time is of order 1/|ln & varepsilon;| and the extinction rate associated with the QSD mu(& varepsilon;) has a lower bound of order 1/|ln & varepsilon;|; (2) If Lambda(0 )> 0, then mu(& varepsilon; )-> mu(0), the mean extinction time is polynomial in 1/& varepsilon;(2 )and the extinction rate is polynomial in & varepsilon;(2). Furthermore, when Lambda(0 )> 0 we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD mu(& varepsilon;) and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in & varepsilon; for the time spent close to mu(& varepsilon;). In contrast to models that only take into account demographic stochasticity, our results demonstrate the significant effect of environmental stochasticity-it turns an exponentially long mean extinction time to a sub-exponential one.