  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 necessary modeling
and simulation tools. Care is also taken to establish a solid mathematical
foundation on which our modeling framework is built. This is the subject of the
first half of the paper. The second half is devoted to developing a multi-scale
framework for eco-evolutionary modeling, where the relevant scales are that of
species and individual/population, respectively. Hence, a system of interacting
species can be described at the species level, while for branching species a
population level description is necessary. Our multi-scale framework thus
consists of coupling the species and population level models where speciation
events are detected in advance and then resolved at the population scale until
the branching is complete. Moreover, since the population level model is
formulated as a PDE, we first establish the well-posedness in the time-discrete
setting, and then derive the a posteriori error estimates which provides a
fully computable upper bound on an energy-type error, including also for the
case of general smooth distributions (which will be useful for the detection of
speciation events). Several numerical tests validate our framework in practice.
