Persistence of mutualisms with bidirectional interactions in a two-species system

In Lotka-Volterra equations (LVEs) of mutualisms, population densities of mutualists will increase infinitely if the mutualisms between them are strong, which is called the divergence problem. In order to avoid the problem, a mutualism system of two species is analyzed in this work. The model is derived from reactions on lattice and has a form similar to that of LVEs. Population densities of species will not increase infinitely because of spatial limitation on the lattice. Stability analysis of the model demonstrates basic mechanisms by which the mutualisms lead to coexistence/extinction of the species. When in coexistence, intermediate mutualistic effect is shown to lead to the maximal density in certain parameter ranges, while a strong or weak mutualistic effect is not so good. Furthermore, the stability analysis exhibits that extremely strong/weak mutualisms will result in extinction of one/both species.