Title

A mean field model for species abundance

Date of Award

1999

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

J. Theodore Cox

Keywords

Ewens sampling, Mean field, Species abundance

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

In this thesis, we use the multitype mean field voter model as a model of species interaction, to obtain results about species abundance. Briefly, we start with the complete graph on n vertices, K n , with each site occupied by a particle. Particles are represented by a value in (0, 1), where distinct values represent different species. Particles, then undergo mutation at rate α, and are relabeled with a value chosen uniformly from (0, 1). Particles also give birth at rate 1, and invade any of the other n sites randomly. This process has a unique stationary distribution denoted by [Special characters omitted.] , which is given by the Ewens sampling formula. For each value in (0, 1) that is present in [Special characters omitted.] , we count the number particles represented by the same value, and call that the patch size of the species. Let K n [ a, b ] denote the number of species with patch size in [ a, b ]. We study the limiting distribution of K n [ a, b ] as the mutation rate α tends to 0, which will in turn force [Special characters omitted.] . In particular, we obtain results about [Special characters omitted.] , and [Special characters omitted.] , where [Special characters omitted.] .

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