Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the asymptotic limi
t, but also the propagator. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anti-correlated Levy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.
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 w
alker 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 s
ystem 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.
We introduce a model of traveling agents ({it e.g.} frugivorous animals) who feed on randomly located vegetation patches and disperse their seeds, thus modifying the spatial distribution of resources in the long term. It is assumed that the survival
probability of a seed increases with the distance to the parent patch and decreases with the size of the colonized patch. In turn, the foraging agents use a deterministic strategy with memory, that makes them visit the largest possible patches accessible within minimal travelling distances. The combination of these interactions produce complex spatio-temporal patterns. If the patches have a small initial size, the vegetation total mass (biomass) increases with time and reaches a maximum corresponding to a self-organized critical state with power-law distributed patch sizes and Levy-like movement patterns for the foragers. However, this state collapses as the biomass sharply decreases to reach a noisy stationary regime characterized by corrections to scaling. In systems with low plant competition, the efficiency of the foraging rules leads to the formation of heterogeneous vegetation patterns with $1/f^{alpha}$ frequency spectra, and contributes, rather counter-intuitively, to lower the biomass levels.
Many studies on animal and human movement patterns report the existence of scaling laws and power-law distributions. Whereas a number of random walk models have been proposed to explain observations, in many situations individuals actually rely on me
ntal maps to explore strongly heterogeneous environments. In this work we study a model of a deterministic walker, visiting sites randomly distributed on the plane and with varying weight or attractiveness. At each step, the walker minimizes a function that depends on the distance to the next unvisited target (cost) and on the weight of that target (gain). If the target weight distribution is a power-law, $p(k)sim k^{-beta}$, in some range of the exponent $beta$, the foraging medium induces movements that are similar to Levy flights and are characterized by non-trivial exponents. We explore variations of the choice rule in order to test the robustness of the model and argue that the addition of noise has a limited impact on the dynamics in strongly disordered media.
We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length sc
ale growing in time as $t^{1/2}$. Further from onset, the characteristic lengths along the preferred directions $hat{x}$ and $hat{y}$ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and Model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results of electroconvection in nematics is presented.
We investigate the properties of a deterministic walk, whose locomotion rule is always to travel to the nearest site. Initially the sites are randomly distributed in a closed rectangular ($A/L times L)$ landscape and, once reached, they become unavai
lable for future visits. As expected, the walker step lengths present characteristic scales in one ($L to 0$) and two ($A/L sim L$) dimensions. However, we find scale invariance for an intermediate geometry, when the landscape is a thin strip-like region. This result is induced geometrically by a dynamical trapping mechanism, leading to a power law distribution for the step lengths. The relevance of our findings in broader contexts -- of both deterministic and random walks -- is also briefly discussed.
