Asymptotic harvesting of populations in random environments

TL;DR

This paper applies ergodic optimal control to determine sustainable harvesting strategies in stochastic populations, revealing threshold-based policies and how the optimal approach varies with the yield function's shape.

Contribution

It introduces ergodic optimal control to population harvesting, providing new insights into optimal strategies and their dependence on population dynamics and yield functions.

Findings

Optimal harvesting converges to a unique invariant measure.

Bang-bang strategy with a threshold exists for identity yield function.

Optimal control type depends on the concavity or convexity of the yield function.

Abstract

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction -- instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold $x^*>0$ such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is $C^2$ and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.