Quasistationarity and extinction for population processes under asymptotic reversibility conditions
Year: 2025
Type: article
Abstract: We consider stochastic population processes that are almost surely absorbed at the origin within finite time. Our interest is in the quasistationary distribution, $${\varvec{u}}$$ u , and the expected time, $$\tau $$ τ , from quasistationarity to extinction, both of which we study via WKB approximation. This approach involves solving a Hamilton-Jacobi partial differential equation specific to the model. We provide conditions under which analytical solution of the Hamilton-Jacobi equation is possible, and give the solution. This provides a first approximation to both $${\varvec{u}}$$ u and $$\tau $$ τ . We provide further conditions under which a corresponding ‘transport equation’ may be solved, leading to an improved approximation of $${\varvec{u}}$$ u . For multitype birth and death processes, we then consider an alternative approximation for $${\varvec{u}}$$ u that is valid close to the origin, provide conditions under which the elements of this alternative approximation may be found explicitly, and hence derive an improved approximation for $$\tau $$ τ . We illustrate our results in a number of applications.
Source: Journal of Mathematical Biology
Author: Damian Clancy
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4415463086