Optimal escape-and-feeding dynamics of random walkers: Rethinking the convenience of ballistic strategies

Excited random walks represent a convenient model to study food intake in a media which is progressively depleted by the walker. Trajectories in the model alternate between (i) feeding and (ii) escape (when food is missed and so it must be found again) periods, each governed by different movement rules. Here, we explore the case where the escape dynamics is adaptive, so at short times an area-restricted search is carried out, and a switch to extensive or ballistic motion occurs later if necessary. We derive for this case explicit analytical expressions of the mean escape time and the asymptotic growth of the depleted region in one dimension. These, together with numerical results in two dimensions, provide surprising evidence that ballistic searches are detrimental in such scenarios, a result which could explain why ballistic movement is barely observed in animal searches at microscopic and millimetric scales, therefore providing significant implications for biological foraging.

Figure 1

Schematic picture of the ERW dynamics considered. (a) The 1D domain is initially full of food (upper panel) and is depleted through successive feeding periods at speed $v_f$ separated by escape periods. (b) Time dynamics of $l\left(t\right)$, which grows uniformly for feeding (marked as $f$) and stays constant during the escape (marked as the $e$ phase).

Figure 2

Nondimensional mean escape time as a function of the switching rate $w$ for different values of the ratio $x_0/l$ as follows: (1) $x_0/l={10}^{-1}$, (2) $x_0/l={10}^{-2}$, (3) $x_0/l={10}^{-3}$, (4) $x_0/l={10}^{-4}$. In all cases we use $l=100, v_e={\alpha }_1=\gamma =1$. Inset: Survival probability of the walker in the escape phase for a constant mortality rate $r=v_e/l$. The values of the parameters are the same as those used for the main figure.

Figure 3

Mean size of the depleted region as a function of time for different values of the switching rate $w$, as follows: circles ($w=10$), triangles ($w=1$), inverted triangles ($w=0.1$), diamonds ($w=0.01$), pentagons ($w=0.001$). The dotted lines correspond to the corresponding asymptotic expression derived in (18). Inset: Mean food intake relative to the ballistic escape case for long times ($t={10}^5$ for solid symbols, $t={10}^7$ for open symbols) as a function of $w$. In all cases we use $v_f=v_e={\alpha }_1=\gamma =1$.

Figure 4

Food intake $\langle N\rangle$ in two dimensions (as defined in the main text) as a function of time for different values of the switching rate $w$, as follows: circles ($w=10$), triangles ($w=1$), inverted triangles ($w=0.1$), diamonds ($w=0.01$), pentagons ($w=0.001$). Inset: Mean food consumed at long times relative to the ballistic escape case, for long times ($t={10}^5$ for solid symbols, $t={10}^7$ for open symbols) as a function of $w$. In all cases we use $v_f=v_e={\alpha }_1=\gamma =1$.