[Paper Review] Optimal sustainable harvesting of populations in random environments

H R E Luis Alvarez, Alexandru Hening|arXiv (Cornell University)|Jul 6, 2018

Stochastic processes and statistical mechanics24 references4 citations

TL;DR

This paper proposes an optimal sustainable harvesting strategy for populations in random environments by maximizing both the expected and almost sure asymptotic yield. It proves the existence of a unique optimal threshold policy—using local time push—under weak assumptions, with explicit characterization in cases including the stochastic Verhulst-Pearl model.

ABSTRACT

We study the optimal sustainable harvesting of a population that lives in a random environment. The novelty of our setting is that we maximize the asymptotic harvesting yield, both in an expected value and almost sure sense, for a large class of harvesting strategies and unstructured population models. We prove under relatively weak assumptions that there exists a unique optimal harvesting strategy characterized by an optimal threshold below which the population is maintained at all times by utilizing a local time push-type policy. We also discuss, through Abelian limits, how our results are related to the optimal harvesting strategies when one maximizes the expected cumulative present value of the harvesting yield and establish a simple connection and ordering between the values and optimal boundaries. Finally, we explicitly characterize the optimal harvesting strategies in two different cases, one of which is the celebrated stochastic Verhulst Pearl logistic model of population growth.

Motivation & Objective

• To develop a sustainable harvesting strategy that maximizes long-term yield in randomly fluctuating environments.
• To establish the existence and uniqueness of an optimal harvesting policy under weak assumptions on population dynamics.
• To characterize the optimal strategy as a local time push-type policy that maintains population above a critical threshold.
• To connect the asymptotic yield maximization framework with the classical expected cumulative present value approach via Abelian limits.
• To explicitly derive optimal strategies for specific models, including the stochastic Verhulst-Pearl logistic model.

Proposed method

• Formulates a general class of unstructured population models subject to environmental noise.
• Defines the asymptotic yield as the long-run average harvest, optimizing both in expectation and almost surely.
• Applies stochastic control theory to derive the optimal policy, showing it is characterized by a threshold below which the population is protected via local time push.
• Uses dynamic programming and variational inequalities to establish optimality conditions.
• Applies Abelian limit arguments to relate the asymptotic yield framework to the expected cumulative present value criterion.
• Solves explicitly for two cases: the general diffusion model and the stochastic Verhulst-Pearl logistic model.

Experimental results

Research questions

• RQ1 What is the optimal harvesting strategy that maximizes the long-run average yield in a randomly fluctuating environment?
• RQ2 How does the optimal threshold policy relate to the classical expected cumulative present value maximization framework?
• RQ3 Under what conditions does a unique optimal harvesting strategy exist for general population models?
• RQ4 What is the explicit form of the optimal strategy in the stochastic Verhulst-Pearl logistic model?
• RQ5 How do the values and optimal boundaries compare between the asymptotic yield and cumulative present value optimization criteria?

Key findings

• There exists a unique optimal harvesting strategy that maintains the population above a critical threshold using a local time push policy.
• The optimal strategy ensures both the expected and almost sure asymptotic yield are maximized under weak regularity assumptions.
• The Abelian limit establishes a clear ordering between the optimal boundaries of the asymptotic yield and cumulative present value problems.
• The optimal threshold in the asymptotic yield framework is strictly smaller than that in the cumulative present value framework.
• For the stochastic Verhulst-Pearl model, the optimal harvesting strategy is explicitly characterized as a threshold policy with a specific critical population level.
• The results demonstrate a robust connection between long-term sustainability and dynamic control in stochastic population models.