Darwinian evolutionary distributions with time-delays

Evolutionary distributions (ED) refer to the solution of systems of reaction diffusion equations that mimic Darwinian evolution. In their general form, ED-dynamics comprise a set of nonlinear partial differential equations (PDE). To each ED, there belongs a separate adaptive space with its own dimensionality in which phenotypes evolve by mutations (diffusion) along heritable adaptive traits. The mutations are small and random.\ They occur at birth. The separate adaptive spaces are linked through the reaction terms; not through the mutation terms. Some of the systems we investigate cannot lead to Turing instability: they are either stable or not stable only. ED-dynamics with delays indicate that in classical reaction-diffusion models of predator-prey-to the extent that they imitate such systems in Nature-Turing instability is not likely to arise. When Turing instability occurs, it is within a narrow range of parameter values. Thus, patterns in the distribution of phenotypes-as a consequence of evolution by natural selection-are not likely to be observed in Nature.