Interspecific Segregation in a Lattice Ecosystem with Intraspecific Competition

Abstract

Coexistence of competing species is extremely common in nature. If competing species live separately, then they can coexist easily. In the present article, we present a lattice version of Lotka–Volterra competition model. This system contains two competing species. They can coexist, when intraspecific competition is stronger than interspecific competition. We find a self-organized isolation of microhabitat: when the densities of both species approach zero, living regions of both species are naturally and completely separated from each other. Previously, co-workers in our laboratory presented a model of habitat segregation. In this case, however, a specific interaction was assumed: the rock-paper-scissors relation was assumed among three species. In contrast, the model presented here may be applicable to many ecosystems, since we only assume competition which is very common in real systems. Our model composed of two species X and Y is defined by j! E ðrate: mjÞ ð1aÞ jþ E! 2j ðrate: rjÞ ð1bÞ jþ j! jþ E ðrate: cjÞ ð1cÞ Here j is the lattice site occupied by species j ( j 1⁄4 X, Y), and E is the empty site. The pair of reactions (1a) and (1b) are called the contact process, where (1a) and (1b) represent death and birth processes, respectively. The last reaction means the intraspecific competition. An example is the direct competition between a pair of plants; neighboring plants are competing to take sunlight or resources. The parameters mj, rj and cj respectively denote mortality, reproduction and competition rates. Besides the direct competition (1c), our model contains a tacit (indirect) competition: according to reaction (1b), any pair of individuals compete to get the empty space E. Simulations are performed by the method of lattice Lotka–Volterra model: 1) Initially, we distribute X and Y on the square lattice. 2) The lattice is updated as follows: (i) we perform two-body reactions (1b) and (1c). Choose one lattice site randomly, and then randomly specify one of four neighboring sites. Let the pair react; for example, if the pair of sites are (X, E), then E is changed into X with the rate rx. (ii) we perform one-body reactions (1a). Choose one lattice point randomly; if the site is X (or Y), it will become E by the rate mx (or my). 3) Repeat step 2) until the system reaches a stationary state. If the global interaction is allowed between any pair of lattice sites, the population dynamics of our system (1) is given by the mean-field theory (rate equation):