## The Annals of Probability

### A spatial model for the abundance of species

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855647

Digital Object Identifier
doi:10.1214/aop/1022855647

Mathematical Reviews number (MathSciNet)
MR1626495

Zentralblatt MATH identifier
0935.60100

#### Citation

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. https://projecteuclid.org/euclid.aop/1022855647

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