Abstract
Optimal random foraging strategy has gained increasing attention. It is shown that L{\'e}vy flight is more efficient compared with the Brownian motion when the targets are sparse. However, standard L{\'e}vy flight generally cannot be followed in practice. In this paper, we assume that each flight of the forager is possibly interrupted by some uncertain factors, such as obstacles on the flight direction, natural enemies in the vision distance, and restrictions in the energy storage for each flight, and introduce the tempered L{\'e}vy distribution $p(l)\sim {\rm e}^{-\rho l}l^{-\mu}$. It is validated by both theoretical analyses and simulation results that a higher searching efficiency can be achieved when a smaller $\rho$ or $\mu$ is chosen. Moreover, by taking the flight time as the waiting time, the master equation of the random searching procedure can be obtained. Interestingly, we build two different types of master equations: one is the standard diffusion equation and the other one is the tempered fractional diffusion equation.
Highlights
One common approach to the animal movement patterns is to use the scheme of optimizing random search [1,2,3]
In a random search model, single or multiple individuals search a landscape to find targets whose locations are not known a priori, which is usually adopted to describe the scenario of animals foraging for food, prey or resources
The locomotion of the individual has a certain degree of freedom which is characterized by a specific search strategy such as a type of random walk and is subject to other external or internal constraints, such as the environmental context of the landscape or the physical and psychological conditions of the individual
Summary
One common approach to the animal movement patterns is to use the scheme of optimizing random search [1,2,3]. The foraging behavior of the wandering albatross on the ocean surface was found to obey a power law distribution [8]; the foraging patterns of a free-ranging spider monkey in the forests was found to be a power law tailed distribution of steps consistent with Lévy walks [9, 10] On this basis, researchers mainly consider two issues: one is to model the foraging behavior as a Lévy flight and the other one is to study the searching efficiency theoretically or experimentally. While the searching efficiency is higher when μ tends to 1 for the destructive case (the target site found by the forager becomes undetectable in subsequent flights). For more details about the model and existing results, one may refer to the works of Viswanathan et al [6, 7] and references therein
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