Dynamics of a lattice gas model

Lotka-Volterra equations (LVEs) for mutualisms predict that when mutualistic effects between species are strong, population sizes of the species increase infinitely, which is the so-called divergence problem. Although many models have been established to avoid the problem, most of them are rather complicated. This paper considers a mutualism model of two species, which is derived from reactions on lattice and has a form similar to that of LVEs. Population sizes in the model will not increase infinitely since there is interspecific competition for sites on the lattice. Global dynamics of the model demonstrate essential features of mutualisms and basic mechanisms by which the mutualisms can lead to persistence/extinction of mutualists. Our analysis not only confirms typical dynamics obtained by numerical simulations in a previous work, but also exhibits a new one. Saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation in the system are demonstrated, while a relationship between saddle-node bifurcation and pitchfork bifurcation in the model is displayed. Numerical simulations validate and extend our conclusions.