Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights

Individual animal movement has been a focus of intense research and considerable controversy over the last two decades, however the understanding of wider ecological implications of various movement behaviours is lacking. In this paper, we consider this issue in the context of pattern formation. Using an individual-based modelling approach and computer simulations, we first show that density dependence ("auto-taxis") of the individual movement in a population of random walkers typically results in the formation of a strongly heterogeneous population distribution consisting of clearly defined animals clusters or patches. We then show that, when the movement takes place in a large spatial domain, the properties of the clusters are significantly different in the populations of Brownian and non-Brownian walkers. Whilst clusters tend to be stable in the case of Brownian motion, in the population of Levy walkers clusters are dynamical so that the number of clusters fluctuates in the course of time. We also show that the population dynamics of non-Brownian walkers exhibits two different time scales: a short time scale of the relaxation of the initial condition and a long time scale when one type of dynamics is replaced by another. Finally, we show that the distribution of sample values in the populations of Brownian and non-Brownian walkers is significantly different.