Reinforced diffusions as models of memory-mediated animal movement

Abstract

How memory shapes animals' movement paths is a topic of growing interest in ecology, with connections to planning for conservation and climate change. Empirical studies suggest that memory has both temporal and spatial components, and can include both attractive and aversive elements. Here, we introduce reinforced diffusions (the continuous time counterpart of reinforced random walks) as a modelling framework for understanding the role that memory plays in determining animal movements. This framework includes reinforcement via functions of time before present and of distance away from a current location. Focusing on the interplay between memory and central place attraction (a component of home ranging behaviour), we explore patterns of space usage that result from the reinforced diffusion. Our efforts identify three qualitatively different behaviours: bounded wandering behaviour that does not collapse spatially, collapse to a very small area, and, most intriguingly, convergence to a cycle. Subsequent applications show how reinforced diffusion can create movement trajectories emulating the learning of movement routes by homing pigeons and consolidation of ant travel paths. The mathematically explicit manner with which assumptions about the structure of memory can be stated and subsequently explored provides linkages to biological concepts like an animal's ‘immediate surroundings' and memory decay.

Power-law relaxation of a confined diffusing particle subject to resetting with memory

“…In a more applied context, random walks with long range memory have become increasingly useful in ecology for the description and analysis of animal mobility. There is mounting evidence that animals do not follow pure Markov processes but use their memory and tend to revisit preferred places during ranging [11][12][13][14][15][16]. The model above was able to describe quantitatively the movement patterns of Capuchin monkeys in the wild [1], as well as of individual elks released in an unknown environment [17].…”

Lévy movements and a slowly decaying memory allow efficient collective learning in groups of interacting foragers

“…The use of memory, as well as its decay over time, can be inferred by fitting models to real trajectories obtained from tracking devices in foraging landscapes where the resource patches are well identified [27,28]. Recent theoretical approaches incorporating memory-based movements into ordinaryrandom-walk models have shown how spatial learning can emerge in principle [29][30][31]. This phenomenon is noticeable by frequent revisits to certain places where resources are located [32] or through the emergence of home ranges and preferred travel routes [33,34].…”