Fractional Herd Behavior with Cross-Diffusion

Survival behaviors among marine biological populations have been studied by quite many researchers [270, 395]. For example, diffusion as the most significant phenomenon can be regarded as the diffusion of standardized random walks linearly with time. But the life activities remain complicated. Predators in the close-range scale can adjust their foraging with near-end clues by using the information of resource distribution [393]. Furthermore, predators mainly apply their foraging strategies to get the foraging success in a wide range of mesoscale like from several kilometers to hundreds of kilometers [394]. However, within the mesoscale boundary, there exists no uniform foraging activities for some predators such as a big divergent between the typical random walk theory and the statistic feature of their real data. So that to study the spatial scale dynamically becomes important and valuable for understanding the related biological behavior. In addition, some experiment evidence proves that there occurs abnormal diffusion in nature [377, 379, 387]. L $$\acute{e}$$ e ´ vy flight model, as one of the most widely effective models for these abnormal diffusion phenomena, has been applied for the foraging phenomena by marine predators empirically. For instance, based on the ocean remote sensing data, it is noted that predation process of penguins, the sharks, and other ocean creatures can be modeled effectively by the L $$\acute{e}$$ e ´ vy flying model [394] not the original random walk model. According to the belief that L $$\acute{e}$$ e ´ vy flight model can be used commonly for open ocean foragers as diffusion behavior, a fractional-order model is introduced in the diffusion process.