Extinction times for a birth–death process with weak competition
We consider a birth–death process with birth rates iλ and death rates iμ+i(i−1)θ, where i is the current state of the process. A positive competition rate θ is assumed to be small. In the supercritical case where λ > μ, this process can be viewed as a demographic model for a population with high carrying capacity around (λ−μ)/θ. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as θ → 0. All three reproduction regimes λ > μ, λ < μ, and λ = μ are studied.
Keywordsbirth–death process carrying capacity time to extinction coupling method logistic branching process
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- 5.G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, 3rd ed., Oxford Univ. Press, New York, 2001.Google Scholar