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Adaptation to an environment consisting of two patches (each with different optimal strategy) is investigated. The patches have independent density regulation ('soft selection'). If the patches are similar enough and migration between them is strong, then evolution ends up with a generalist ESS. If either the difference between the patches increases or migration weakens, then the generalist strategy represents a branching singularity: The initially monomorphic population first evolves towards the generalist strategy, there it undergoes branching, and finally two specialist strategies form an evolutionarily stable coalition. Further increasing the between-patch difference or decreasing migration causes the generalist to lose its convergence stability as well, and an initially monomorphic population evolves towards one of the specialists optimally adapted to one of the two patches. Bifurcation pattern of the singularities is presented as a function of patch difference and migration rate. Connection to speciation theory is discussed. The transition from the generalist ESS to the coexisting pair of specialist strategies is regarded as a clonal prototype of parapatric (if the between-patch difference increases) or allopatric (if the migration decreases) speciation. We conclude that the geographic and the competitive speciation modes are not distinct classes.

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We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the singular points of the evolutionary state space. Here "evolutionary branching" may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.