Size inequality and the tragedy of the commons phenomenon in plant competition

Game theory predicts that the evolutionarily stable level of root production is greater for plants grown with neighbours compared to plants grown alone, even when the available resources per plant are constant. This follows from the fact that for plants grown alone, new roots compete only with other roots on the same plant, whereas for multiple plants grown in a group, new roots can also compete with the roots of other plants, thereby potentially acquiring otherwise unavailable resources at their neighbours' expense. This phenomenon, which results in plants grown with neighbours over-proliferating roots at the expense of above-ground biomass, has been described as a 'tragedy of the commons', and requires that plants can distinguish self from non-self tissues. While this game theoretical model predicts the evolutionarily stable strategies of individual plants, it has only been tested on average allocation patterns of groups of plants. This is problematic, because average patterns can appear to reflect a tragedy of the commons, even when none has occurred. In particular, assuming (1) a decelerating relationship between individual plant biomass and the amount of resources available, and (2) greater size inequality in plants grown with neighbours compared to plants grown alone (due to asymmetric competition), then plants grown with neighbours should, at least on average, be smaller than plants grown alone. This is a manifestation of 'Jensen's Inequality', which states that for decelerating functions, the average value of the function is less than the function of the average value. We suggest that Jensen's Inequality should serve as an appropriate null hypothesis for examining biologically-based explanations of changes in biomass allocation strategies.