From individual to collective displacements in heterogeneous environments

Animal displacement plays a central role in many ecological questions. It can be interpreted as a combination of components that only depend on the animal (for example a random walk) and external influences given by the heterogeneity of the environment. Here we treat the case where animals switch between random walks in a homogeneous 2D environment and its 1D boundary, combined with a tendency for wall-following behaviour (thigmotactism) that is treated as a Markovian process. In the first part we use mesoscopic techniques to derive from these assumptions a set of partial differential equations (PDE) with specific boundary conditions and parameters that are directly given by the individual displacement parameters. All assumptions and approximations made during this derivation are rigorously validated for the case of exploratory behaviour of the ant Messor sanctus. These PDE predict that the stationary density ratio between the 2D (centre) and 1D (border) environment only depends on the thigmotactic component, not on the size of the centre or border areas. In the second part we test this prediction with the same exploratory behaviour of M. sanctus, in particular when many ants move around simultaneously and may interact directly or indirectly. The prediction holds when there is a low degree of heterogeneity (simple square arena with straight borders), the collective behaviour is “simply” the sum of the individual behaviours. But this prediction breaks down when heterogeneity increases (obstacles inside the arena) due to the emergence of pheromone trails. Our approach may be applied to study the effects of animal displacement in any environment where the animals are confronted with an alternation of 2D space and 1D borders as for example in fragmented landscapes.