## Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Optimal resource allocation in the plant-fungi mutualism for a growing system
Kouki Uchinomiya, Yoh Iwasa

#### Abstract

Many terrestrial plants have mutualistic relation with soil fungi. The plant supplies carbon obtained by photosynthesis to the fungi. The fungi also allocate some of the soil nutrition such as phosphorus to the plant. This is important for the survivorship and the growth rate of a small plant (i.e. seedling). By using mathematical method, we discuss the optimal allocation of resources by each player (plant and fungus) when the whole system is growing exponentially.

In this study we assume one plant-one fungus system. In addition they can get more resource when they are grown. When plant gets carbon, plant decides how much allocate it to the fungus. On the other hand, fungus allocates phosphorus to the plant. We first analyse the optimal allocation fraction of resources to the opponent that maximize the speed of the growth of the whole system. As a result, there are 3 types of allocation: allocating all resource, allocating a part of resource and allocating no resource. Allocating all resource is interpreted as investment. By allocating all resource, plant or fungus can encourage the growth of the partner. The partner will give more resource in the future.

Next we analyse the case both plant and fungus allocate a part of resource. We assume they grow exponentially. The results depend on acquisition function of resource. If resource acquisition is based on Cobb-Douglas function, optimal allocation fraction is constant that is independent of partnerâ€™s demand. However, if resource acquisition ability is determined by lesser resource, the resource allocation fraction depends on resource demand of both plant and fungus. When phosphorus is more important for plant, both plant and fungus increase the allocation fraction. On the other hand, carbon is important more important for plant, plant decreases its allocation fraction, but fungus increases its allocation fraction. Reversely, the fungus increases when carbon is important and decreases when phosphorus is important. The differences of the results depending on the function suggest that if resource acquisition is based on power function, the resource allocation does not change when interact with other partners. However, if resource acquisition ability is determined by lesser resource, the resource allocation changes depending on the partner. These results indicate that the difference of resource acquisition may cause the difference of resource allocation strategy.

#### Keywords

Mutualism; Mycorrhiza; Resource allocaiton