Biological population management based on a Hamilton–Jacobi–Bellman equation with boundary blow up

Abstract

Establishment of a long-term biological population management policy requires balancing its cost and utility/disutility. We approach this issue from the viewpoint of stochastic control as an efficient candidate for its modelling and analysis. A remarkable point of the present model is making good use of an unbounded performance index, which naturally penalises too much population decrease. The goal of the present optimal control problem is approached through solving a Hamilton–Jacobi–Bellman equation having a solution blowing up at a boundary. We thus positively use a singular mathematical model. Solution behaviour of the HJB equation is analysed from the viewpoint of viscosity solutions with the help of an asymptotic expansion technique, which can handle the blow up under regularity assumptions. Numerical framework for analysing the optimal control problem is presented and examined as well, to qualitatively determine the profile of the solution, optimal human intervention, and probability density function.

KEYWORDS:

• Biological population dynamics
• stochastic control
• viscosity solution
• boundary singularity
• numerical discretisation

Acknowledgements

JSPS Research grant number 17K15345 supports this research. The comments from the two reviewers significantly improved the contents of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science [grant number 17K15345].