Searching on patch networks using correlated random walks: Space usage and optimal foraging predictions using Markov chain models

We describe a novel representation of a discrete correlated random walk as the transition matrix of a Markov chain with the displacements as the states. Such a representation makes it possible to utilize results from the theory of absorbing Markov chains, to make biologically interesting predictions without having to resort to Monte Carlo simulations. Our motivation for constructing such a representation is to explore the relationship between the movement strategy of an animal searching for resources upon a network of patches, and its consequent utilization of space and foraging success. As an illustrative case study, we have determined the optimal movement strategy and the consequent usage of space for a central place forager utilizing a continuous movement space which is discretized as a hexagonal lattice. The optimal movement strategy determines the size of the optimal home range. In this example, the animal uses mnemokinesis, which is a sinuosity regulating mechanism, to return it to the central place. The movement strategy thus refers to the choice of the intrinsic path sinuosity and the strength of the mnemokinetic mechanism. Although the movement space has been discretized as a regular lattice in this example, the method can be readily applied to naturally compartmentalized movement spaces, such as forest canopy networks. This paper is thus an attempt at incorporating results from the theory of random walk-based animal movements into Foraging Theory.