On the number of births and deaths during an extinction cycle, and the survival of a certain individual in a competition process

Abstract

Competition processes, as discussed by Iglehart (1964) [26] and Reuter (1961) [25], have been frequently used in biology to describe the dynamics of population models involving some kind of interaction among various species. Our interest is in the stochastic model of a competition process analyzed by Ridler-Rowe (1978) [23], which is related to an ecosystem of two species. The ecosystem is closed in the sense that no immigration or emigration is supposed to take place. Individuals compete either directly or indirectly for common resources and, consequently, births and deaths depend on the population sizes of one or both of the species. In this paper, we focus on the number of births and deaths during an extinction cycle. Specifically, we discuss an approximation method inspired from the use of the maximum size distribution, which is equally applicable to the survival of a certain individual. We analyze three models defined in terms of the way individuals within each species are selected to die. Our results are illustrated with reference to simulated data.