Asymptotic harvesting of populations in random environments

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 [Formula: see text] 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 [Formula: see text] 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.

Optimal sustainable harvesting of populations in random environments

“…We are interested (as in [HNUW18]) in the maximization of the expected asymptotic harvesting yield (also called the expected average cumulative yield ) of the population. Before presenting our main findings on the optimal ergodic harvesting strategy and the maximal expected average cumulative yield we first establish the following auxiliary verification lemma.…”

“…of harvested individuals. As in [HNUW18], and in contrast to what happens in a significant part of the literature (see [LES94,LES95,AES98,LØ97]), the optimal strategy will be such that the population is never depleted and cannot be harvested to extinction. This is clear since if Z T → 0 in some sense then ℓ = 0 in the above equation.…”

“…We build on the results from [AES98, AE01] and [HNUW18]. Suppose that in an infinitesimal time dt we harvest a quantity dZ t where (Z t ) t≥0 is any adapted, non-negative, non-decreasing, and right continuous process.…”

“…Our model does not involve any discount factors. We generalize the setting of [HNUW18] where the authors assumed the harvesting rate was bounded by some parameter M > 0. This corresponds to having total control over the harvested population.…”

Modeling stochastic operation of reservoir under ambiguity with an emphasis on river management

“…The long‐time limit considers behavior of Φ for t → +∞ where . Here, is referred to as the effective Hamiltonian, which is a value function under the temporal averaging in ergodic control . The limit equation for t → +∞ is where the left‐hand side is independent from t .…”