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 study both power-law decay and exponential decay of the smell with the distance of the food from the forager. For power-law decay, we vary the exponent α governing this decay, while for exponential decay we vary λ also governing the rate of the decay. For power-law decay we find, both analytically and through simulations, that for a forager living in one dimension, there is a critical value of α, namely α_c, where for α < α_c the forager will die in finite time, however for α > α_c the forager has a nonzero probability to live infinite time. We calculate analytically the critical value, α_c, separating these two behaviors and find that α_c depends on S as α_c=1+1/⌈S/2⌉. We find analytically that at α = α_c the system has an essential singularity. For exponential decay we find analytically that for all λ, the forager has a finite probability to live for infinite time. 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 α (and λ) for which the forager has the longest lifetime.