The dynamics of evolutionary branching in an ecological model

Abstract

Eco-evolutionary modelling involves the coupling of ecological equations to evolutionary ones. The interaction between ecological dynamics and evolutionary processes is essential to simulating evolutionary branching, a precursor to speciation. The creation and maintenance of biodiversity in models depends upon their ability to capture the dynamics of evolutionary branching. Understanding these systems requires low-dimension models that are amenable to analysis. The rapid reproduction rates of marine plankton ecosystems and their importance in determining the fluxes of climatically important gases between the ocean and atmosphere suggest that the next generation of global climate models needs to incorporate eco-evolutionary models in the ocean. This requires simple population-level models, that can represent such eco-evolutionary processes with orders of magnitude fewer equations than models that follow the dynamics of individual phenotypes. We present a general framework for developing eco-evolutionary models and consider its general properties. This framework defines a fitness function and assumes a beta distribution of phenotype abundances within each population. It simulates the change in total population size, the mean trait value, and the trait differentiation, from which the variance of trait values in the population may be calculated. We test the efficacy of the eco-evolutionary modelling framework by comparing the dynamics of evolutionary branching in a six-equation eco-evolutionary model that has evolutionary branching, with that of an equivalent one-hundred equation model that simulates the dynamics of every phenotype in the population. The latter model does not involve a population fitness function, nor does it assume a distribution of phenotype abundance across trait values. The eco-evolutionary population model and the phenotype model produce similar evolutionary branching, both qualitatively and quantitatively, in both symmetric and asymmetric fitness landscapes. In order to better understand the six-equation model, we develop a heuristic three-equation eco-evolutionary model. We use the density-independent mortality parameter as a convenient bifurcation parameter, so that differences in evolutionary branching dynamics in symmetric and asymmetric fitness landscapes may be investigated. This model shows that evolutionary branching of a stable population is flagged by a zero in the local trait curvature; the trait curvature then changes sign from negative to positive and back to negative, along the solution. It suggests that evolutionary branching points may be generated differently, with different dynamical properties, depending upon, in this case, the symmetry of the system. It also suggests that a changing environment, that may change attributes such as mortality, could have profound effects on an ecosystem’s ability to adapt. Our results suggest that the properties of the three-dimensional model can provide useful insights into the properties of the higher-dimension models. In particular, the bifurcation properties of the simple model predict the processes by which the more complicated models produce evolutionary branching points. The corresponding bifurcation properties of the phenotype and population models, evident in the dynamics of the phenotype distributions they predict, suggest that our eco-evolutionary modelling framework captures the essential properties that underlie the evolution of phenotypes in populations.