Forager with intermittent rest: Better for survival?

Abstract

We study the fate of a forager who searches for food performing a random walk on lattices. The forager consumes the available food on the site it visits and leaves it depleted but can survive up to $S$ steps without food. We introduce the concept of intermittent rest in the dynamics which allows the forager to rest with probability $p$ upon consumption of food. The parameter $p$ significantly affects the lifetime of the forager, showing that the intermittent rest can be beneficial for the forager for chosen parameter values. The study of various other quantities reveals interesting scaling behavior with $p$ and also departure from usual diffusive behavior for $0.5 < p < 1$. In addition to numerical simulations, the problem has been studied with analytical approach in one dimension and the results up to $p < 0.5$ agree with the numerical ones to a large extent.

Figures (11)

• Figure 1: (a) Lifetime $\tau$ against starvation time $S$ for $0 \leqslant p < 1$ in 1$d$. Power law behavior is observed for $S \gg S_{\text{crossing}}$ as $\tau \sim \kappa S^{\Delta}$, with $\Delta \simeq 1.0$ and $\kappa \simeq 3.18$. In the inset, the behavior for the scaled lifetime $\tau/S^{\Delta}$ is shown. The deviation from the conventional scaling is observed over a larger range of values of $S$ as $p$ increases and its maximum value occurs at $S = S_{\text{max}}$, with $S_{\text{max}}$ increasing with $p$. (b) $S_{\text{crossing}}$ is plotted against $1-p$ and the curve is dictated by $S_{\text{crossing}} = \frac{A}{1-p}$, where $A = 1.64 \pm 0.03$. (c) $S_{\text{max}}$ as a function of $1-p$ is shown; $S_{\text{max}} = \frac{B}{1-p}$ with $B=7.98 \pm 0.12$.
• Figure 2: (a) Lifetime $\tau$ against starvation time $S$ with $0 \leqslant p < 1$ in $2d$. Similar power law behavior as in Fig. 1 (a) is observed with the exponent $\Delta \simeq 1.9$ and $\kappa \simeq 0.91$. (b) $S_{\text{crossing}}$ against $1 - p$. It has been observed that $S_{\text{crossing}} = \frac{A}{1 - p}$ where $A = 3.52 \pm 0.19$.(c) $S_{\text{max}}$ as a function of $1 - p$. $S_{\text{max}} = \frac{B}{(1 - p)^z}$ with $B = 8.60 \pm 0.32$ and $z \simeq 1.19$.
• Figure 3: Behavior of average number of distinct site $N$ as a function of starvation time $S$ with $0 \leqslant p < 1$. For $S \gg S_{\text{merge}}$, power law behavior is observed, the exponent being $\xi \simeq 0.50$. $S_{\text{merge}}$ against $p$ has been shown in the inset, which follows a relation $S_{\text{merge}} = \frac{c}{1-p}$ where, $c =10.26 \pm 0.11$.
• Figure 4: Average number of distinct site $N$ against starvation time $S$ in $2d$ with $0 \leqslant p < 1$. For $S \gg S_{\text{merge}}$, $\xi \simeq 1.78$. $S_{\text{merge}} = \frac{c}{1-p}$ is shown in the inset with $c = 25.91 \pm 0.84$.
• Figure 5: Variation of lifetime $\tau$ with average number of distinct site $N$ for $1d$. The variation follows the relation $\tau(N) = \alpha(p) N$ in small $N$ region for $p<1$. In the large $N$ region, the curve goes as $N^2$. In the inset, the variation of $\alpha$ with $1-p$ is shown as $\alpha = \frac{1}{1-p}$.