  We study a non-Markovian and nonstationary model of animal mobility
incorporating both exploration and memory in the form of preferential returns.
We derive exact results for the probability of visiting a given number of sites
and develop a practical WKB approximation to treat the nonstationary problem.
We further show that this model adequately describes empirical movement data of
Egyptian fruit bats (Rousettus aegyptiacus) when accounting for
inter-individual variation in the population. Finally, we study the probability
of visiting any site a given number of times and derive the corresponding
mean-field equation. Here, we find a remarkable phase transition occurring at
preferential returns which scale linearly with past visits. Following empirical
evidence, we suggest that this phase transition reflects a trade-off between
extensive and intensive foraging modes.
