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On the stability of a population growth model with sexual reproduction on \(\mathbb{Z}^ d\), \(d\geq 2\). (English) Zbl 0814.60100

Summary: We continue our study on the stability properties of a population growth model with sexual reproduction on \(\mathbb{Z}^ d\), \(d \geq 2\). In the author’s previous work [ibid. 20, No. 1, 232-285 (1992; Zbl 0752.60087)], it was proved that in the type IV process (the two-dimensional symmetric model on \(\mathbb{Z}^ 2)\), the vacant state \(\emptyset\) 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). We prove that in the type III process on \(\mathbb{Z}^ 2\), the vacant state \(\emptyset\) is stable under the second kind of perturbation, and in three- or higher-dimensional symmetric models, the vacant state \(\emptyset\) 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0752.60087
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