Adaptive dynamic resource allocation can cause tragedy of the commons in plants with nutrient competition

Plants exhibit plastic responses to the absence or presence of competitors. When competing for soil nutrients, plants often show root overproliferation compared to when they grow without competitors. This excessive investment in roots to acquire more nutrients can reduce reproductive yield (e.g., seed mass), a phenomenon known as the tragedy of the commons (TOC). The mechanisms of this phenomenon have been investigated theoretically, focusing on resource allocation strategies between the aboveground (shoot) and the belowground (roots) parts. The previous studies have primarily considered these strategies in terms of sizes of those parts or static allocation rates to those over the season, overlooking dynamic change of allocation within the season. In this study, we introduced a concept of dynamic resource allocation into the plant competition game and investigate the optimal resource allocation strategy using Pontryagin's maximum principle. Based on the solutions of schedules, we explored the mechanism causing TOC in nutrient competition. Our findings reveal that plants adopt the singular control (i.e., simultaneous allocation to shoot and root), where the control trajectory is identical regardless of the presence or absence of competitors, although the period of simultaneous allocation become longer in the presence of competitors. This trend associates with increasing the root size and decreasing the shoot size at the end of season in the competitive case. Our analysis demonstrates that TOC in plant nutrient competition arises from differences in the allocation period to roots in the competitive scenario.