Abstract
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ℝ+, and reproduce independently new particles according to a probability law p(ξ n ) on ℕ. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
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This work was supported by the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012) and Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
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Li, Y., Liu, Q. Age-dependent branching processes in random environments. Sci. China Ser. A-Math. 51, 1807–1830 (2008). https://doi.org/10.1007/s11425-008-0065-4
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DOI: https://doi.org/10.1007/s11425-008-0065-4
Keywords
- age-dependent branching processes
- random environments
- probability generating function
- integral equation
- extinction probability
- exponential growth rates of expectation and conditional expectation
- random walks and renewal equation in random environments
- renewal theorem