The Annals of Probability

A spatial model for the abundance of species

Maury Bramson, J. Theodore Cox, and Richard Durrett

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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 $\alpha \to \infty$, 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.

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Ann. Probab. Volume 26, Number 2 (1998), 658-709.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D25: Population dynamics (general)

Species abundance distributions multitype voter model coalescing random walk


Bramson, Maury; Cox, J. Theodore; Durrett, Richard. A spatial model for the abundance of species. Ann. Probab. 26 (1998), no. 2, 658--709. doi:10.1214/aop/1022855647.

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