Optimal allocation ratios: A square root relationship between the ratios of symbiotic costs and benefits

All organisms struggle to make sense of environmental stimuli in order to maximize their fitness. For animals, single cells and superorganisms responses to stimuli are generally proportional to stimulus ratios – a phenomenon described by Weber’s Law. However, Weber’s Law has not yet been used to predict how plants respond to stimuli generated from their symbiotic partners. Here, we develop a model for quantitatively predicting the carbon (C) allocation ratios into symbionts that provide nutrients to their plant host. Consistent with Weber’s Law, our model demonstrates the optimal ratio of resources allocated into a less- relative to the more-beneficial symbiont scale to the ratio of the growth benefits of the two strains. As C allocation into symbionts increases, the ratio of C allocation into two strains approaches the square root of the ratio of symbiotic growth benefits (e.g., a worse symbiont providing ¼ the benefits gets sqrt(¼) =1/2 the C of a better symbiont). We document a compe...