Optimum resource allocation in the plant-fungus symbiosis for an exponentially growing system

Question: What is the optimal allocation of carbon by a plant to its mycorrhizal fungus, and what is the optimal allocation of phosphorus by the mycorrhizal fungus to the plant? Method: We examined the level of resource allocation by both a plant and fungus (hereafter, 'players') that achieved the most rapid growth within the overall system during the exponential growth phase. We assumed resource acquisition rates of the Cobb-Douglas type. We also analysed the dynamic optimization condition (singular subarc) for each player to maximize its own fitness given its partner's growth schedule. Results: Each player should allocate more to its partner when the resource provided by the partner is more important to its own resource acquisition. The ESS resource allocation by each player is equivalent to that which achieves the cooperative optimum.