## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 4 (1980), 727-744.

### On Some Two-Sex Population Models

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 4 (1980), 727-744.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994662

**Digital Object Identifier**

doi:10.1214/aop/1176994662

**Mathematical Reviews number (MathSciNet)**

MR577312

**Zentralblatt MATH identifier**

0442.60080

**JSTOR**

links.jstor.org

**Subjects**

Primary: 92A15

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]

**Keywords**

Population model problem of the sexes marriage function Malthusian parameter reproductive value deterministic differential equation pure birth process almost sure convergence central limit theorem law of the iterated logarithm moment expansion

#### Citation

Asmussen, Soren. On Some Two-Sex Population Models. Ann. Probab. 8 (1980), no. 4, 727--744. doi:10.1214/aop/1176994662. https://projecteuclid.org/euclid.aop/1176994662