Evolution of predators and prey kills Turing patterns

The spatiotemporal patterns of predators and their prey play a pivotal role in ecology and ecological interactions can drive their formation at fine scales. While motility can explain the emergence of such predator-prey patterns via the Turing mechanism, the predicted Turing patterns do not exhibit temporal changes that are common in experiments and nature. Moreover, the Turing mechanism treats motility as fixed, even though predators and prey adjust their motility in response to each other and their interactions influence their evolution. Using adaptive dynamics, I prove that the evolution of motility prevents the formation of Turing patterns and promotes the formation of dynamic patterns, such as predator-prey waves. The resulting predator-prey cycles are shown to be induced by heterogeneous motility, which extends the emergence of predator-prey cycles beyond regimes predicted by Lotka-Volterra or Rosenzweig-MacArthur models. This work unites models for predator-prey spatiotemporal patterns and evolution of motility to explain how dynamic spatiotemporal patterns of co-evolving predators and prey emerge and persist. The novel mathematical theory is general and extends to other ecological situations, such as ecological public goods games.