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August, 1980 On Some Two-Sex Population Models
Soren Asmussen
Ann. Probab. 8(4): 727-744 (August, 1980). DOI: 10.1214/aop/1176994662


Let $M_t$ be the number of males and $F_t$ the number of females present at time $t$ in a population where births take place at rates which at time $t$ are $mR(M_t, F_t)$ and $fR(M_t, F_t)$ for males and females, respectively. Assume that $R$ has the form $R(M, F) = (M + F)h(M/(M + F))$ with $h$ sufficiently smooth at $m/(m + f)$. A Malthusian parameter $\lambda$ and a random variable $W$ such that $e^{-\lambda t}M_t\rightarrow mW, e^{-\lambda}F_t\rightarrow fW$ a.s. are exhibited, the rate of convergence is found in form of a central limit theorem and a law of the iterated logarithm and an asymptotic expansion of the reproductive value function $\tilde{V}(M, F) = E(W\mid M_0 = M, F_0 = F)$ is given. Also some discussion of an associated set of deterministic differential equations is offered and the stochastic model compared to the solutions.


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Soren Asmussen. "On Some Two-Sex Population Models." Ann. Probab. 8 (4) 727 - 744, August, 1980.


Published: August, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0442.60080
MathSciNet: MR577312
Digital Object Identifier: 10.1214/aop/1176994662

Primary: 92A15
Secondary: 60J80 , 60J85

Keywords: Almost sure convergence , central limit theorem , deterministic differential equation , Law of the iterated logarithm , Malthusian parameter , marriage function , moment expansion , population model , problem of the sexes , pure birth process , reproductive value

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 4 • August, 1980
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