The Physics of Foraging

Abstract

Do the movements of animals, including humans, follow patterns that can be described quantitatively by simple laws of motion? If so, then why? These questions have attracted the attention of scientists in many disciplines, and stimulated debates ranging from ecological matters to queries such as 'how can there be free will if one follows a law of motion?' This is the first book on this rapidly evolving subject, introducing random searches and foraging in a way that can be understood by readers without a previous background on the subject. It reviews theory as well as experiment, addresses open problems and perspectives, and discusses applications ranging from the colonization of Madagascar by Austronesians to the diffusion of genetically modified crops. The book will interest physicists working in the field of anomalous diffusion and movement ecology as well as ecologists already familiar with the concepts and methods of statistical physics.

Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

“…The hallmark of an anomalous diffusive process is such that the mean squared displacement is proportional to t 2H with H = 1 2 , [58,73,74]. Theoretical investigations instigated by Scher and Montroll [58] in their description of dispersive transport in amorphous semi-conductors, found that traditional methods based on Brownian models failed.…”

“…We aim to address these questions and elaborate on current studies where it has been shown that in the case of a genuine Lévy walk, the problem of trap count interpretation can still be addressed based on the diffusion equation [1,50]. Our base study, exploits the fact that the Lévy parameter 1 < μ < 3 corresponds to the super-diffusive regime [73]. The underlying issue is, what properties must be inherent in D(t) so that trap counts for Lévy random walkers can be predicted, effectively.…”

“…This property is widely regarded as a 'finger-print' of Brownian motion [12,46,50,71,73]. More generally, if D = D(t) is explicitly dependent on time, then (3.1) can be transformed by introducing…”

Anomalous Fluctuation Relations

“…[70]. Lévy flights define one of the most paradigmatic models of anomalous dynamics with wide applications, for example, in fluid dynamics [71], in the foraging of biological organisms [72] and in glassy optical material [44], to highlight only a few cases. It can be shown that the position PDF ̺(x, t) characterizing the process defined by…”

“…There is an ongoing debate about whether biological organisms such as, e.g., albatrosses, marine predators and fruit flies have managed to mimimize the search time for food in a way that matches to optimizing search strategies in terms of stochastic processes; see Refs. [72,95] and further references therein. In particular, it has been argued that Lévy flights are superior to Brownian motion in order to find sparsely, randomly distributed, replenishing food sources [95].…”