Collective dynamics of territorial animals
Collective movement patterns appear at all scales from microorganisms to invertebrate and vertebrate animals. In certain cases the individual entities communicate indirectly modifying the environment in which they roam by leaving a trace of their action or passage. This form of interaction has a long tradition in the ecological literature and is called stigmergy. In the context of territorial mammals modification of the environment occurs because of scent deposition and is being exploited to maintain exclusive ownership of certain region of space. By introducing the so-called territorial random walkers, it is possible to study the formation of territorial patterns by modelling the movement and interaction of scent-depositing animals. Territorial random walkers consist of agents that move at random and deposit scent, that is mark the locations they visit using temporal flags that decay over a finite amount of time, and retreat upon encountering a foreign scent. Depending solely on the ratio between the time for which the mark is active and the time it takes for the walker to cover its own territory, the system displays different patterns. Short lived marks produce rapidly morphing, fast traveling territories. A broad range of shapes and territory sizes are observed, and these territories may display ergodic trajectories. Marks that remain active for long times yield slowly moving territories that resemble glassy systems. In such state territories are effectively confined in space and have a more homogeneous shape distribution. I will show how these different regimes emerge based on the population density and the length of time for which marks remain active. I will also present an adiabatic mean-field approximation that allows to describe at short times the dynamics of the walker and that of the territory boundaries through a Fokker-Planck formalism.