Multispecies Lottery Competition: A Diffusion Analysis

Abstract

The lottery model is a stochastic competition model designed for space-limited communities of sedentary organisms. Examples of such communities may include coral reef fishes (Chesson & Warner 1981), aquatic sessile organisms (Fagerstrom 1988), and plant communities such as trees in a tropical forest (Leigh 1982; Hatfield et al., in press). The lottery model, and its properties and behavior, has been discussed previously (Chesson & Warner 1981; Chesson 1982, 1984, 1991, 1994; Warner & Chesson 1985; Chesson & Huntly 1988). Furthermore, explicit conditions for the coexistence of two species and the stationary distribution of the two-species model were determined (in Hatfield & Chesson 1989) using an approximation with a diffusion process (Karlin & Taylor 1981). However, a diffusion approximation for the multispecies model (for more than two species) has not been reported previously, and a stagestructured version has not been investigated. The stage-structured lottery model would be more reasonable for communities of long-lived species in which recruitment or death rates depend on the age or stage of the individuals (e.g., trees in a forest). In this chapter, we present a diffusion approximation for the multispecies lottery model and also discuss a stage-structured version of this model.