Scaling mobility patterns and collective movements: deterministic walks in lattice Han Xiao-Pu Zhou Tao Wang Bing-Hong University of Science and Technology.

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Scaling mobility patterns and collective movements: deterministic walks in lattice Han Xiao-Pu Zhou Tao Wang Bing-Hong University of Science and Technology of China

Empirical results of biological mobility patterns Previous explanations of such property Our model Conclusions

Scaling biological mobility patterns Spider monkey Behav. Ecol. Sociobiol. (2004) 55:223, Levy walk patterns in the foraging movements of spider monkeys PNAS (2003) 100 :12771, Helical Levy walks: Adjusting searching statistics to resource availability in microzooplankton Microzooplankton

Empirical results: marine predator Nature_451_1098-Scaling laws of marine predator search behaviour

Human mobility patterns Nature (2008) 453:779, Understanding individual human mobility patterns Nature (2006) 439:462, The scaling laws of human travel

Explanation: Optimizing searches Nature_401_911-Optimizing the success of random searches

Dynamical model: Deterministic walks PhysRevE_75_ Origin of power-law distributions in deterministic walks- The influence of landscape geometry PhysA_342_329-Modeling the searching behavior of social monkeys

Our model Start point: seeking advantage least action Based on deterministic walks

Basic Rules One or several walkers moving on 2D lattices The resource of each position can slowly recover Walkers exhaust the resource of their occupied position, and jump to the nearest and richest position

Main Parameters M: number of walkers, denotes the group size N: size of lattice r = MV/N^2, denotes the degree of resource richness V: the maximum value of the resource in a lattice

Typical trajectories M = 100 N = 500 Move 5000 steps

Moving length distribution

Collective Movements Just like the marching bands of locust Two movies

The periodic evolution of ordering parameter

Evolution of averaged ordering parameter The absence of food leads to the collective movements

Notice There is not any directly interactions between individuals Different with most of the models for collective movements

Approximately analysis

Conclusions Our model starts from two universal properties of biological behavior Our model can generate power-law flight length distribution with exponent -1.6 to -3.0, in agreement with empirical results Around the critical point, our model generates ordered collective movements of walkers with a quasi-periodic synchronization of walkers' directions. Our findings provide a bridge to connect the individual scaling mobility patterns and the ordered collective movements