Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.
Non-motile elongated bacteria confined in two-dimensional open micro-channels can exhibit collective motion and form dense monolayers with nematic order if the cells proliferate, i.e., grow and divide. Using soft molecular dynamics simulations of a system of rods interacting through short range mechanical forces, we study the effects of the cell growth rate, the cell aspect ratio and of the sliding friction on nematic ordering and on pressure fluctuations in confined environments. Our results indicate that rods with aspect ratio >3.0 reach quasi-perfect nematic states at low sliding friction. At higher frictions, the global nematic order parameter shows intermittent fluctuations due to sudden losses of order and the time intervals between these bursts are power-law distributed. The pressure transverse to the channel axis can vary abruptly in time and shows hysteresis due to lateral crowding effects. The longitudinal pressure field is on average correlated to nematic order, but it is locally very heterogeneous and its distribution follows an inverse power-law, in sharp contrast with non-active granular systems. We discuss some implications of these findings for tissue growth.
Many attempts to relate animal foraging patterns to landscape heterogeneity are focused on the analysis of foragers movements. Resource detection patterns in space and time are not commonly studied, yet they are tightly coupled to landscape properties and add relevant information on foraging behavior. By exploring simple foraging models in unpredictable environments we show that the distribution of intervals between detected prey (detection statistics)is mostly determined by the spatial structure of the prey field and essentially distinct from predator displacement statistics. Detections are expected to be Poissonian in uniform random environments for markedly different foraging movements (e.g. Levy and ballistic). This prediction is supported by data on the time intervals between diving events on short-range foraging seabirds such as the thick-billed murre ({it Uria lomvia}). However, Poissonian detection statistics is not observed in long-range seabirds such as the wandering albatross ({it Diomedea exulans}) due to the fractal nature of the prey field, covering a wide range of spatial scales. For this scenario, models of fractal prey fields induce non-Poissonian patterns of detection in good agreement with two albatross data sets. We find that the specific shape of the distribution of time intervals between prey detection is mainly driven by meso and submeso-scale landscape structures and depends little on the forager strategy or behavioral responses.
