Population dynamics under demographic and environmental stochasticity

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 $\frac{1}{\epsilon^{2}}$ for small $\epsilon>0$ (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) $\mu_\epsilon$ 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, i.e., when $\epsilon\to 0$. We show that the external Lyapunov exponent still plays a key role: 1) If $\Lambda_0<0$, then $\mu_\epsilon\to \delta_0$, the mean extinction time is of order $|\ln \epsilon|$ and the extinction rate associated with the QSD $\mu_{\epsilon}$ has a lower bound of order $\frac{1}{|\ln\epsilon|}$; 2) If $\Lambda_0>0$, then $\mu_\epsilon\to \mu_0$, the mean extinction time is polynomial in $\frac{1}{\epsilon^{2}}$ and the extinction rate is polynomial in $\epsilon^{2}$. Furthermore, when $\Lambda_0>0$ we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD $\mu_\epsilon$ and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in $\epsilon$ for the time spent close to $\mu_\epsilon$.