Lifetime of a greedy forager with long-range smell

We study a greedy forager who consumes food throughout a region. If the forager does not eat any food for $S$ time steps it dies. We assume that the forager moves preferentially in the direction of greatest smell of food. Each food item in a given direction contributes towards the total smell of food in that direction, however the smell of any individual food item decays with its distance from the forager. We assume a power-law decay of the smell with the distance of the food from the forager and vary the exponent $\alpha$ governing this decay. We find, both analytically and through simulations, that for a forager living in one dimension, there is a critical value of $\alpha$, namely $\alpha_c$, where for $\alpha<\alpha_c$ the forager will die in finite time, however for $\alpha>\alpha_c$ the forager has a nonzero probability to live infinite time. We calculate analytically, the critical value, $\alpha_c$, separating these two behaviors and find that $\alpha_c$ depends on $S$ as $\alpha_c=1 + 1/\lceil S/2 \rceil$. We determine analytically that at $\alpha=\alpha_c$ the system has an essential singularity. We also study, using simulations, a forager with long-range decaying smell in two dimensions (2D) and find that for this case the forager always dies within finite time. However, in 2D we observe indications of an optimal $\alpha$ for which the forager has the longest lifetime.