Nonstationary dynamics of encounters: Mean valuable territory covered by a random searcher

Inspired by recent experiments on the organism Caenorhabditis elegans we present a stochastic problem to capture the adaptive dynamics of search in living beings, which involves the exploration-exploitation dilemma between remaining in a previously preferred area and relocating to new places. We assess the question of search efficiency by introducing a new magnitude, the mean valuable territory covered by a Browinan searcher, for the case where each site in the domain becomes valuable only after a random time controlled by a nonhomogeneous rate which expands from the origin outwards. We explore analytically this magnitude for domains of dimensions 1, 2, and 3 and discuss the theoretical and applied (biological) interest of our approach. As the main results here, we (i) report the existence of some universal scaling properties for the mean valuable territory covered as a function of time and (ii) reveal the emergence of an optimal diffusivity which appears only for domains in two and higher dimensions.

Figure 1

(a) Individual C. elegans trajectory obtained after passing it from a nutrient-rich to a nutrient-poor environment, with the starting point defined at $(0,0).$ (Experimental data generated at William S. Ryu Lab, University of Toronto, Canada.) (b–e) Snapshots at increasing times [from (b) to (e)] of a single realization of the stochastic process described in the text. Red lines represent the walker trajectory. Sites on the lattice become valuable with time (shown in gray). Sites in black are those visited after they become valuable, so the objective of the walker should be to maximize the black region.

Figure 2

Mean valuable territory covered by a Brownian walker up to time $t$ for different values of $d$ and $\alpha$, with (a) $d=1,\alpha =0$, (b) $d=1,\alpha =1$, (c) $d=2,\alpha =0$, and (d) $d=2, \alpha =1$. Symbols correspond to the numerical results obtained for ${\beta }_{\alpha }$: ${10}^{-2}$ (open circles), ${10}^{-3}$ (filled circles), ${10}^{-4}$ (open triangles), and ${10}^{-5}$ (filled triangles). Solid lines showing the asymptotic scaling and the new ($\alpha$-dependent) intermediate scaling reported here are provided as a visual guide.

Figure 3

Mean valuable territory covered by a Brownian walker up to a characteristic leaving time $\tau$ for different $\alpha$ values in (a) one, (b) two, and (c) three dimensions. Symbols correspond to random-walk simulations, with $\alpha =1$ (open circles), $\alpha =2$ (filled circles), and $\alpha =3$ (triangles). Solid lines correspond to the numerical computation of $V_{\tau }$ from (6), while dotted and dashed lines correspond to approximations in Eqs. (8)–(10) and (11)–(13), respectively. Insets: Relative error found by the approximations (with $\mathrm{\Delta }{V}_{\tau }$ the absolute value of the difference between the approximation and the exact solution); again, dotted and dashed lines correspond to the approximations in Eqs. (8)–(10) and (11)–(13). Values used in all cases are $b=0.25, D=1$, and $\tau =50$ (one dimension) or $\tau =200$ (two and three dimensions).

Figure 4

$V_{\tau }$ as a function of the nondimensional parameter $\eta$ [computed from (6)]. Inset: Optimal values of $\eta$ maximizing $V_{\tau }$. In all cases shown we have used parameter values $b=0.25, D=1, {\beta }_{\alpha }=0.001$, and $\tau =200$.