Microbial cross-feeding promotes multiple stable states and species coexistence, but also susceptibility to cheaters.

Mutualism, interspecific cooperation that yields reciprocal benefits, can promote species coexistence, enhancing biodiversity. As a specific form of mutualism, cross-feeding, where each of two mutualists produces a resource the other one needs, has been broadly studied. However, few theoretical studies have examined competition between cross-feeding mutualists and cheaters, who do not synthesize resources themselves. In this paper we study a model with two mutualists, a cheater, two micronutrients that are synthesized and exchanged by the mutualists, and one macronutrient that is only from external supply. We investigate the coexistence of the species in the framework of resource competition theory. In particular, we examine the effect of the mutualists' synthesis rates on their coexistence. In the absence of cheaters, multiple stable states occur if the synthesis rates are high, and higher synthesis rates increase the possibility that mutualists coexist. However, when the cheater is present, higher synthesis rates promote invasion by the cheater: If the cheater is superior on all three resources, it will either persist with at most one mutualist or even trigger extinction of all three species; if the cheater is only superior on the macronutrient, both mutualists may still coexist with the cheater. Our results provide a framework for further study on more complex mutualistic networks and real microbial communities.