  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.
