## The Annals of Probability

### On the Stability of a Population Growth Model with Sexual Reproduction on $Z^d, d \geq 2$

Hwa-Nien Chen

#### Abstract

We continue our study on the stability properties of a population growth model with sexual reproduction on $\mathbf{Z}^d, d \geq 2$. In the author's previous work, it was proved that in the type IV process (the two-dimensional symmetric model on $\mathbf{Z}^2$), the vacant state $\varnothing$ is stable under perturbation of the initial state (the first kind of perturbation), and it is unstable under perturbation of the birth rate (the second kind of perturbation). In this paper we prove that in the type III process on $\mathbf{Z}^2$, the vacant state $\varnothing$ is stable under the second kind of perturbation, and in three or higher-dimensional symmetric models, the vacant state $\varnothing$ is unstable under the first kind of perturbation. These results, combined with the results obtained earlier, provide a fairly complete picture concerning the stability properties of these models.

#### Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1195-1226.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988600

Digital Object Identifier
doi:10.1214/aop/1176988600

Mathematical Reviews number (MathSciNet)
MR1303642

Zentralblatt MATH identifier
0814.60100

JSTOR
Chen, Hwa-Nien. On the Stability of a Population Growth Model with Sexual Reproduction on $Z^d, d \geq 2$. Ann. Probab. 22 (1994), no. 3, 1195--1226. doi:10.1214/aop/1176988600. https://projecteuclid.org/euclid.aop/1176988600