Population Dynamic Models Generating Species Abundance Distributions of the Gamma Type

This paper deals with processes generating species abundance distributions of the gamma type, including Fisherʹs logarithmic series model, MacArthurʹs broken stick model and the extended gamma model. Speciation is described by a Poisson process and density-dependence within species is given by the logistic growth function. Environmental fluctuations are modelled by constant environmental variances in specific rates of population growth. All models are based on diffusion approximations for the abundance of each species. The simplest case of independent identically distributed abundances is generalized in two ways. Interspecific competition is introduced in a way that makes the total abundance for the community constant. The stationary distribution for the corresponding multivariate diffusion for the relative abundances is the Dirichlet distribution, though with a slight reduction in the shape parameter compared to the case of independent abundances. Heterogeneity between the species is discussed in general and exemplified by two types of mixing in Fisherʹs model leading to Kemptonʹs generalized log-series model.