Evolutionary Stagnation Due to Pattern-Pattern Interactions in a Coevolutionary Predator-Prey Model

We consider a spatially stmctured model of a coevolutionary predator-prey system with interactions in a one-dimensional phenotype space. We show that in phenotype space predators and prey organize themselves into distinct clusters of phenotypes called quasi-species. The prey quasi-species also cluster in patches in real space. As the prey quasi-species evolve away from the predator quasi-species (in phenotype space) the prey patch size reduces and the single predator quasi-species is inhibited from evolving toward either of the two prey species. We show that it is the interaction between the phenotype space patterns (quasi-species) and the real space patterns (patches) that inhibit the predators from evolving.