Does greed help a forager survive?

We investigate the role of greed on the lifetime of a random-walking forager on an initially resource-rich lattice. Whenever the forager lands on a food-containing site, all the food there is eaten and the forager can hop $\mathcal{S}$ more steps without food before starving. Upon reaching an empty site, the forager comes one time unit closer to starvation. The forager is also greedy—given a choice to move to an empty or to a food-containing site in its local neighborhood, the forager moves preferentially toward food. Surprisingly, the forager lifetime varies nonmonotonically with greed, with different senses of the nonmonotonicity in one and two dimensions. Also unexpectedly, the forager lifetime in one dimension has a huge peak for very negative greed where the forager is food averse.

Figure 1

Greedy forager motion in $d=1$ and $d=2$. Solid and open circles indicate food and empty sites, respectively. Arrow widths indicate relative hopping probabilities.

Figure 2

Dependence of the scaled forager lifetime $\mathcal{T}/\mathcal{S}$ on greediness $G$ in $d=1$ (a) and $d=2$ (b). The inset compares simulations with the analytic result (12) for $G$ close to $-1$ and $\mathcal{S}={10}^6$. Dotted curves in $d=2$ correspond to a nonbacktracking walk (see summary text).

Figure 3

Schematic illustration of the space-time trajectory of a greedy forager in the semi-infinite geometry. The shaded region denotes food.

Figure 4

A random-walk trajectory that leads to trapping of a perfectly greedy forager. (a) Forager ($\times$) at the decision point. (b) Forager hops to the interior region (shaded). (c) Food in the interior is completely consumed, so that the forager ($\times$) may be trapped inside the newly created desert.

Figure 5

Hopping probabilities for a greedy forager inside a desert of length $L$.