A cooperative system of two species with bidirectional interactions.

Cooperation between species is often regarded to mean that the increase of each species promotes the growth of the other. The well-known cooperative model is the Lotka-Volterra equations (LVEs). In the LVEs, population densities of species increase infinitely as the cooperation is strong, which is called the divergence problem. Moreover, LVEs never exhibit an Allee effect in the case of obligate cooperation. In order to avoid these problems, several models have been established although most of them are rather complex. In this paper, we consider a cooperative system of two species with bidirectional interactions, in which each species also has negative feedback on the other. Population densities of the species will not increase infinitely because of the limited resource and negative feedback. Then, we focus on an extended lattice model of cooperation, which is deduced from reactions on lattice and has the same form as that of LVEs. In the case of obligate cooperation, the model predicts an Allee effect. Global dynamics of the system exhibit essential features of cooperation and basic mechanisms by which the cooperation can lead to coexistence/extinction of species. Intermediate cooperation is shown to be beneficial in cooperation under certain conditions, while extremely strong cooperation is demonstrated to lead to extinction of one/both species. Numerical simulations confirm and extend our results.