Modeling the process of speciation using a multi-scale framework including error estimates

This paper concerns the modeling and numerical simulation of the process of speciation. In particular, given conditions for which one or more speciation events within an ecosystem occur, our aim is to develop the modeling and simulation tools necessary to handle such events. The model we employ can be considered as an extension of the classical Lotka-Volterra model, where in addition to the species abundance, the model also governs the evolution of the species mean traits and species trait covariances, and in this sense generalizes the purely ecological Lotka-Volterra model to an eco-evolutionary model. Although the model thus allows for evolving species, by construction, it is not able to handle speciation events due to a breakdown of the underlying assumptions on which the model is derived (i.e., unambiguous identification of distinct species in the fitness landscape). Thus, the fundamental problem to overcome regarding speciation is that the unit of species is too coarse to capture the fine-scale dynamics of a speciation event. Our strategy is twofold; monitoring the species in the system and detecting speciation in advance, and, splitting the diverging `parent'-species into new `child'-species. Since the problem at hand is related to that of separate scales, it is convenient to have access to a fine-scale description of the same eco-evolutionary system. For this, we employ a trait-specific population density model governing the dynamics of the abundance density as a function of evolutionary traits. At this scale there is no a priori definition of species, but both species and speciation may be defined a posteriori as e.g., local maxima and saddle points of the population density, respectively.