Optimal control of root nodulation – Prediction of life history theory of a mutualistic system

•Legumes develop root nodules for keeping nitrogen-fixing rhizobial bacteria.•We study the plant's optimal number of nodules considering the cost and benefit.•The optimal nodule number depends on the light at leaves and the soil nitrate.•Dilution effect explains the difference between growing plant and stationary plant. Legumes produce root nodules containing symbiotic rhizobial bacteria that convert atmospheric molecular nitrogen into ammonia or related nitrogenous compounds. The host plant supplies photosynthetic products to root nodules forming a mutualistic system. Legumes have physiological mechanisms for regulating nodule production with chemical signals produced in leaves, called the autoregulation of nodulation. In this paper, we discuss the optimal number of root nodules that maximizes the performance of the host plant. Here, we study two models. In the stationary plant model, the acquired photosynthetic products minus cost and loss are used for reproduction. In the growing plant model, the excess material is invested to produce leaves, roots, and root nodules, resulting in the exponential growth of the whole plant. The analysis shows that having root nodules is beneficial to the plant for a high leaf nitrogen content, faster plant growth rate, a short leaf longevity, a low root/shoot ratio, and low soil nutrient concentration. We discuss the long-distance control of nodulation-autoregulation and dependence on the environmental conditions of terrestrial plants considering these results.