Phase Transition in a Non-Markovian Animal Exploration Model with Preferential Returns

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. Exact results for the probability of visiting a given number of sites are derived and a practical WKB approximation to treat the nonstationary problem is developed. A mean-field version of this model, first suggested by Song et al., [Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010)] was shown to well describe human movement data. We show that our generalized model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for interindividual variation in the population. We also study the probability of visiting any site a given number of times and derive a mean-field equation. Our analysis yields 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.

Figure 1

The probability $P(n,t)$ for $\beta =1$ and $t=1500$. (a) No variation in $\beta$ ($\sigma =0$). We compare simulations (circles), exact result [black dashed-dotted line, Eq. (4)], WKB approximation [red dashed line, Eq. (7)], and WKB approximation at low energies [blue dashed line, Eq. (8)]. (b) Variability in $\beta$ with $\sigma =0.1$: simulations (circles) are compared with a numerical solution of Eq. (9) (dashed line). Insets show $⟨n⟩$ and ${\sigma }_n^2$ (red and black marks, respectively) versus $t$, compared with theory (dashed lines).

Figure 2

(a) The average frequency of visits to the most visited site $f_1$ versus $\alpha$, for $\beta =1$ (simulations). Each curve corresponds to a given number of visits $t$ (see legend). (b) Semi-log plot of $f_k$ versus $\alpha$ for different sites (see legend), for $\beta =1$ and $t=10^5$.

Figure 3

(a)–(b) The mean (blue marks) and variance (red marks) of the number of sites visited by bats during summer and winter, compared to theoretical prediction (black dashed lines), with fitted values of ${\beta }_0=0.71$ and $\sigma =0.21$ in (a), and ${\beta }_0=0.53$ and $\sigma =0.16$ in (b). These are averaged over an ensemble of 38 (a) and 53 (b) bats. (c)–(d) The mean number of visits $⟨m_i⟩$ to the six most visited sites, $i=1,\dots ,6$ from top to bottom (different colors mark different sites), averaged over the same ensembles as in (a) and (b). Black dashed lines $⟨m_i⟩\sim t$ correspond to the theoretical prediction for $\alpha =1$. (e)–(f) Variance of the number of visits ${\sigma }_{m_i}^2$ to the same six sites, averaged over the same ensembles for summer (e) and winter (f). Black dashed lines ${\sigma }_{m_i}^2\sim t^2$ correspond to the theoretical prediction for $\alpha =1$.