We consider a spatial stochastic version of the classical Lotka-Volterra model with interspecific competition.
The classical model is described by a set of ordinary differential equations, one for each species. Mortality is density dependent, including both intraspecific and interspecific competition. Fecundity may depend on the type of species but is density independent. Depending on the relative strengths of interspecific and intraspecific competition and on the fecundities, the parameter space for the classical model is divided into regions where either coexistence, competitive exclusion or founder control occur.
The spatial version is a continuous time Markov process in which individuals are located on the d-dimensional integer lattice. Their dynamics are described by a set of local rules which have the same components as the classical model.
Our main results for the spatial stochastic version can be summarized as follows. Local competitive interactions between species result in (1) a reduction of the parameter region where coexistence occurs in the classical model, (2) a reduction of the parameter region where founder control occurs in the classical model, and (3) spatial segregation of the two species in parts of the parameter region where the classical model predicts coexistence.
"An explicitly spatial version of the Lotka-Volterra model withinterspecific competition." Ann. Appl. Probab. 9 (4) 1226 - 1259, November 1999. https://doi.org/10.1214/aoap/1029962871