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

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.