From Micro-to-Macro: How the Movement Statistics of Individual Walkers Affect the Formation of Segregated Territories in the Territorial Random Walk Model

Animal territoriality is a widespread phenomena in many vertebrate species. In mammals it is often associated with territorial marking with which individuals make their presence conspicuous to others by leaving trace of their passage, often in the form of deposited scent. A simple interaction mechanism consisting of retreating upon the encounter of a foreign scent is sufficient to observe the emergence of territorial patterns at the population level. With the introduction of the so-called territorial random walk model this local avoidance mechanism coupled with a simple diffusive movement of the individuals has been shown to generate long-lasting patterns of segregation at much larger spatial scales. To shed further light on the micro-to-macro connection of this collective movement model we study how the movement statistics of the individuals affect the formation of the segregated scented territories. We represent individual animals as correlated random walkers and we analyse the spatial ordering of the population as a function of the length of time a scent mark remains active after deposition and as a function of the degree of correlation of the movement steps. For low and intermediate correlation strength we find that territories undergo a liquid-hexatic-solid transition as active scent time is increased. Increased spatial order also appears by increasing the correlation strength but only if well away from the ballistic limit. We ascribe this non-monotonic dependence to the coverage efficiency of the individual walkers mainly controlled by the correlation and the mobility of the territories mainly controlled by the active scent time.