Periodic oscillation and tri-stability in mutualism systems with two consumers

This paper considers mutualistic interactions between two consumers, in which one consumer can consume a resource only by exchange of service for service with the other. By rigorous analysis on the one-resource and two-consumer model with Holling-type I response, we show periodic oscillations and tri-stability in the mutualism system: when their initial densities decrease, the consumers' interaction outcomes would change from coexistence in periodic oscillation, to persistence at a steady state, and to extinction. Under certain conditions, we also show two types of bi-stability in the system: the consumers would change from coexisting in periodic oscillation (resp. at a steady state) to going to extinction when their initial densities decrease. Then we analyze a modified system with Holling-type II response. Based on theoretical analysis and numerical computation, we show that there also exist tri-stability and two types of bi-stability in this system. Moreover, it is shown that varying the degree of obligation can lead to transition of interaction outcomes between coexistence in periodic oscillation (resp. at a steady state) and extinction of both consumers. These results are important in understanding complexity in mutualism.