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

How L\'evy flights triggered by presence of defectors affect evolution of cooperation in spatial games