In a discrete-generation, individual-oriented model of predator-prey interactions that exhibits oscillations, we show that the self-structuring of the populations into spiral waves induces a selection pressure for ever-increasing dispersal distances in both populations. As the dispersal distances increase, the sizes of the spatial patterns increase, until they are too large to fit into the limited space. The patterns are then lost and the predators go extinct. This scenario, is, however, not the only outcome. A second selection pressure induced by the spatial boundary can cause reduction of the dispersal distances. Depending on the relative strengths of the two selection pressures, the predators and prey may speciate to give coexistence between short-dispersing boundary quasi-species and far-dispersing spiral quasi-species. Now, when pattern loss occurs, the predators switch to predating on the boundary prey quasi-species and do not go extinct. Also, if the populations reproduce sexually, local gene flow can inhibit the evolution of increasing dispersal distances, and hence the spatial patterns are not lost. Speciation and coexistence can also occur in the sexually reproducing species.