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A spatial model for the abundance of species. (English) Zbl 0935.60100

Summary: The voter model, with mutations occurring at a positive rate \(\alpha\), has a unique equilibrium distribution. We investigate the logarithms of the relative abundance of species for these distributions in \(d\geq 2\). We show that, as \(\alpha\to 0\), the limiting distribution is right triangular in \(d=2\) and uniform in \(d\geq 3\). We also obtain more detailed results for the histograms that biologists use to estimate the underlying density functions.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D25 Population dynamics (general)
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