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

Abstract

In this paper we study a growth model known as the "contact process with sexual reproduction" on $\mathbf{Z}^2$. We focus on the "symmetric" model in which a "child particle" can be produced at a vacant site whenever a pair of its neighboring sites is occupied by "parent particles". Two kinds of stability of the absorbing state $\phi$ (i.e., the state in which all the sites are vacant) are investigated in this paper. The first kind of stability concerns the behavior of the system when it starts close to the state $\phi$. More explicitly, we consider the system starting with a random configuration in which the sites are occupied independently with occupation probability $p$, where $p$ is a small positive parameter. The system is said to be stable if, for sufficiently small $p$, the probability that a site is occupied approaches 0 as time approaches infinity. The second kind of stability concerns the behavior of the system under the perturbation of adding a small quantity $\beta > 0$ to all the birth rates ("spontaneous birth at rate $\beta$"). In this case, stability means that there is an equilibrium state which is close to $\phi$ when $\beta$ is small. It is proven in this paper that in the symmetric model the state $\phi$ is stable under the first kind of perturbation, but it is unstable under the second kind of perturbation.