The eco-evolutionary modelling of populations and their traits using a measure of trait differentiation

We develop new equations for the eco-evolutionary dynamics of populations and their traits. These equations resolve the change in the phenotypic differentiation within a population, which better estimates how the variance of the trait distribution changes. We note that traits may be bounded, assume they may be described by beta distributions with small variances, and develop a coupled ordinary differential equation system to describe the dynamics of the total population, the mean trait value, and a measure of phenotype differentiation. The variance of the trait in the population is calculated from its mean and the population's phenotype differentiation. We consider an example of two competing plant populations to demonstrate the efficacy of the new approach. Each population may trade-off its growth rate against its susceptibility to direct competition from the other population. We create two models of this system: a population model based on our new eco-evolutionary equations; and a phenotype model, in which the growth or demise of each fraction of each population with a defined phenotype is simulated as it interacts with a shared limiting resource and its competing phenotypes and populations. Comparison of four simulation scenarios reveals excellent agreement between the predicted quantities from both models: total populations, the average trait values, the trait variances, and the degree of phenotypic differentiation within each population. In each of the four scenarios simulated, three of which are initially subject to competitive exclusion in the absence of evolution, the populations adapt to coexist. One population maximises growth and dominates, while the other minimises competitive losses. These simulations suggest that our new eco-evolutionary equations may provide an excellent approximation to phenotype changes in populations.