In reciprocal mutualism, interacting species invest resources in their partners. Individuals benefit from these interactions and need to pay the cost of investment for the partner. In the evolution of mutualism, it is a significant question how the level of investment is coevolutionarily determined and resulting the persistence of mutualism. Theoretically, under the given investment level of partner, individuals prefer to reduce their own investment level, which may result in the breakdown of mutualism. While in reality, interacting species are usually able to maintain a mutualistic relationship. This can be explained by the coevolution of investment levels, by which interacting species regulate their investment strategy according to partner’s. Moreover, it is worth noting that there might be other factors that influence the evolutionary outcome of mutualism. One of the possibilities is suppressive factor such as predation or self-depletion, by which species evolve mutualism to compensate for the losses.
    To address this, we develop a mathematical model that describes the increases in plant and fungal biomass over a growing season, including continuous biomass losses due to aging or predation on tissues. In this model, fungi are assumed to be evenly distributed on the host plant's body, resulting in identical rates of biomass depletion for both species. On the two-dimensional plane of their biomasses, the growth trajectory initially converges quickly from the initial state toward the system’s dominant eigenvector, and subsequently moves along this eigenvector with identical depletion rate. Assuming that the initial biomass lies approximately on the eigenvector, we derive the terminal biomasses of both species. We then define the difference between terminal and initial biomass as the fitness of the species, which is considered a proxy for the expansion rate of space occupation. Furthermore, we assume that plants and fungi disperse closely from their natal habitat to ensure the continuous interaction between two interacting species in later generations. Based on this model, we investigate coevolutionary solutions for both the initial state and the resource allocation strategies for the species in mutualism.
    Our preliminary analysis suggests that the reciprocal mutualism can evolve when the difference between growth rates of two interacting species is relatively small with a high biomass depletion rate. Although the conditions for reaching the optimal allocation strategies simultaneously are limited, this result can provide an understanding of the coevolution of resource allocation strategies in mutualistic interactions.