Evolutionary stability of one-to-many mutualisms

One-to-many mutualisms - interspecific cooperations in which each host individual can potentially interact with multiple symbiont individuals while each symbiont individual can only one host individual - are widely found in nature, while their evolutionary stability has not been explored. It has been often thought that partner choice can stabilize multi-player mutualisms. However, in one-to-many mutualisms partner choice is inevitably asymmetric between hosts and symbionts, which might destabilize the system. Here I develop a simple mathematical model for an obligate one-to-many mutualism, with explicitly considering imperfect ability of symbiont choice by hosts. I fix the trait of hosts and concentrate on the evolutionary dynamics of cooperativeness in symbiont population. Each host chooses a constant number of symbionts from a potential symbiont population, a fraction of which are chosen through preferential choice depending on cooperativeness of the symbionts, while the rest are through random choice. After the association between the host and the symbionts is established, the host offers a constant amount of resource to each associating symbiont. It spends a part of the resource to increase the fitness of the host in proportion to its cooperativeness, and the rest for its own reproduction. I show that pure mutualist population is evolutionarily stable when the fraction of preferential choice c is large and the strength of preferential choice k is small, otherwise mutualists and cheaters coexist. In addition, in the coexistence state the frequency of mutualists increases with c. In contrast, it decreases with k, while the cooperativeness of mutualists increases. The two factors offset against each other, so that the fitness gain of host remains constant.