Population persistence and spatially limited social interaction
We study a lattice structured population model in which the mortality of each individual is affected by the number of nearest neighbors. We examine the case in which mortality decreases with the number of neighboring individuals (beneficial or cooperative interaction) and the case in which mortality increases with the number of neighboring individuals (harmful or competitive interaction). Since the reproduction is possible only to neighboring vacant sites, the effect of local density (or a crowding effect) is naturally included. A similar model was studied by \textit{H. Matsuda} et al. [see Lect. Notes Biomath. 71, 154-161 (1987; Zbl 0642.92011)], who calculated, for example, the invadability condition for altruists into a population dominated by nonaltruists. Here we consider a model including only a single type, unlike in Matsuda et al., but instead we incorporate the migration of individuals, expressed as the movement of an individual to a vacant neighbor leaving the original site empty. Migration in effect causes the exchange of a nearest neighbor pair of occupied and vacant sites. By increasing the migration rate, we can change continuously from a perfectly sessile population, studied by Matsuda et al., through a population with limited dispersal, to a completely mixed population. In addition to pair approximation used by Matsuda et al., we also examined improved pair approximation introduced by \textit{K. Satō} et al. [J. Math. Biol. 32, No. 3, 251-268 (1994; Zbl 0790.92022)], and we further examine the modification of the method to allow a more accurate prediction of the equilibrium abundance.
scientific article; zbMATH DE number 797081