Energy Dynamics and Partial Consumption in Foraging

Abstract

In this work, we consider partial consumption of food by a forager in presence of a threshold energy level. The forager considered here can survive for $S$ steps without food, namely the survival time. The threshold limits the consumption of food in such a way that, the forager will only consume food, whenever its energy is below the threshold $k$. Due to partial consumption of food, a site containing food may not always be fully depleted, which in turn helps in increasing the lifetime of the forager. It has been observed that, in our case, the lifetime always increases with $k/S$, although there is a transition threshold $k^*$ below which the increase of lifetime is rapid and above is low. The transition threshold $k^* \sim \sqrt{S}$. The lifetime $τ$ shows a power law behavior as $τ\sim S^β$. For $k/S=0$, the value of $β$ is $4/3$, it then jumps above $2$ and decreases gradually to $1.84$ with increasing $k/S$. Other important quantities like number of revisits to a site, food statistics etc. have been studied and these also show some interesting scaling behavior. The collection of sites either fully or partially depleted of food after the death of the forager $N_{eat}$ shows a crossover behaviour for $k/S \sim 0.5$.

Figures (8)

• Figure 1: Study of lifetime $\tau$ against threshold energy fraction $k/S$ with $S$ as a parameter. It has been observed that in case of partial food consumption, the lifetime never decrease beyond a certain $k^*/S$.
• Figure 2: Plot of $k^*/S$ as a function of starvation time $S$, showing the scaling $k^* \sim A\sqrt{S}$ with $A = 0.978 \pm 0.008$ The top-right inset (a) shows the variation of the average lifetime $\tau$ with $k/S$ for $S = 1024$, illustrating how $k^*/S$ is determined. The fitting lines below and above $k^*/S$ show exponents $l$ and $h$ respectively. The other left bottom inset (b) shows the variation of the fitting exponents $l$ and $h$ with $S$.
• Figure 3: Lifetime $\tau$ against starving time $S$ with varying $k/S$ values. In asymptotic region of $S$, $\tau = \alpha S^{\beta}$ . In the inset, the variation of $\alpha$ and $\beta$ have been shown against $k/S$; $\alpha = a - \frac{b}{1+\left(\frac{(k/S)}{c}\right)^{n}}$ and $\beta = \beta_{\infty} + \exp\left(-\left(\frac{(k/S)}{\theta}\right)^{m}\right)$.
• Figure 4: Plot of scaled number of revisits $N(x)/S^{\mu}$ against $x/S^{\nu}$ with $S = 32, 64, 128, 256, 512$ and $1024$. In the subplot $(a)$, $(b)$ and $(c)$; $k/S = 0.04, 0.08$ and $0.50$ respectively. The fitted function is $N(x)/S^{\mu} = N_0 \exp{(-\lambda |x/S^{\nu}|)}$. In $(d)$ the variation of $\lambda$ against $k/S$ has shown with the exponent = $-0.22 \pm 0.02$.
• Figure 5: Plot of energy $E$ versus time $t$ for $(a)$$k/S = 1.0$ and $(b)$$k/S=0.08$ respectively. The variations are fitted as $E \propto \exp(-\gamma t)$. In $(c)$, the variation of $\gamma$ against $S$ with $k/S = 1.0$ has been shown for different $k/S$. Here, $\gamma \propto S^{-1.80}$, In $(d)$, the variation of $\gamma$ against $k/S$ with $S = 1024$ has been shown and the observed dependence is $\gamma \propto (k/S)^{-0.24}$.