How Lévy flights triggered by presence of defectors affect evolution of cooperation in spatial games

Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a L\'evy flight. A L\'evy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of L\'evy flights for cooperation has been studied by Antonioni and Tomassini. They showed that L\'evy flights promoted cooperation combined with conditional movements triggered by neighboring defectors. However, the optimal condition for neighboring defectors and how the condition changes by the intensity of L\'evy flights are still unclear. Here, we developed an agent-based model in a square lattice where agents perform L\'evy flights depending on the fraction of neighboring defectors. We systematically studied the relationships among three factors for cooperation: sensitivity to defectors, the intensity of L\'evy flights, and population density. Results of evolutionary simulations showed that moderate sensitivity most promoted cooperation. Then, we found that the shortest movements were best for cooperation when the sensitivity to defectors was high. In contrast, when the sensitivity was low, longer movements were best for cooperation. Thus, L\'evy flights, the balance between short and long jumps, promoted cooperation in any sensitivity, which was confirmed by evolutionary simulations. Finally, as the population density became larger, higher sensitivity was more beneficial for cooperation to evolve. Our study highlights that L\'evy flights are an optimal searching strategy not only for foraging but also for constructing cooperative relationships with others.