Depletion of Resources by a Population of Diffusing Species

TL;DR

This paper models how a population of diffusing species depletes resources, deriving the exact probability distribution of depletion time and analyzing how it depends on various factors such as species number and initial resources.

Contribution

It provides an exact analytical form for the depletion time distribution in a resource consumption model involving diffusing species, advancing understanding of resource depletion dynamics.

Findings

Derived the exact probability density of resource depletion time.

Analyzed the impact of species number and initial resources on depletion.

Discussed future research directions and open problems.

Abstract

Depletion of natural and artificial resources is a fundamental problem and a potential cause of economic crises, ecological catastrophes, and death of living organisms. Understanding the depletion process is crucial for its further control and optimized replenishment of resources. In this paper, we investigate a stock depletion by a population of species that undergo an ordinary diffusion and consume resources upon each encounter with the stock. We derive the exact form of the probability density of the random depletion time, at which the stock is exhausted. The dependence of this distribution on the number of species, the initial amount of resources, and the geometric setting is analyzed. Future perspectives and related open problems are discussed.