Evolutionary distributions: Producer-consumer pattern formation

An evolutionary distribution (ED) is defined as a solution of a system of reaction-diffusion equations that mimic Darwinian evolution. To each ED, there belongs a separate adaptive space in which phenotypes evolve by diffusion along adaptive traits. The diffusion is caused by random mutations. Analysis of a producer-consumer ED in parametric space reveals regions where intricate symmetric patterns emerge. Such patterns emerge from models of evolution by natural selection which may explain why they appear in Nature. The patterns are stationary and stable in time and in the adaptive space. No Turing instability arises for ED systems compared with systems where diffusion is in physical space.