A Stochastic Impulse Control Model for Population Management of Fish-Eating Bird Phalacrocorax Carbo and Its Numerical Computation

  • Yuta YaegashiEmail author
  • Hidekazu Yoshioka
  • Koichi Unami
  • Masayuki Fujihara
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 946)


Feeding damage from a fish-eating bird Phalacrocorax carbo to a fish Plecoglossus altivelis is severe in Japan. A stochastic impulse control model is introduced for finding the cost-effective and ecologically conscious population management policy of the bird. The optimal management policy is of a threshold type; if the population reaches an upper threshold, then taking a countermeasure to immediately reduce the bird to a lower threshold. This optimal policy is found through solving a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). We propose a numerical method for HJBQVIs based on a policy iteration approach. Its accuracy on numerical solutions and the associated free boundaries for the management thresholds of the population, is investigated against an exact solution. The computational results indicate that the proposed numerical scheme can successfully solve the HJBQVI with the first-order computational accuracy. In addition, it is shown that the scheme captures the free boundaries subject to errors smaller than element lengths.


Population management Impulse control Hamilton-Jacobi-Bellman quasi-variational inequality 



This paper is partly funded by grants-in-aid for scientific research No.16KT0018, No.17J09125, and No.17K15345 from the Japan Society for the Promotion of Science (JSPS).


  1. 1.
    Natsumeda, T., Sakano, H., Tsuruta, T., Kameda, K., Iguchi, K.I.: Immigration of the common cormorant Phalacrocorax carbo hanedae into inland areas of the northern part of Nagano Prefecture, eastern Japan, inferred from stable isotopes of carbon, nitrogen and sulphur. Fish. Sci. 81(1), 131–137 (2011)CrossRefGoogle Scholar
  2. 2.
    Kameda, K.: Population increase of the Great Cormorant Phalacrocorax carbo and measures to reduce its damage to the fisheries and forests of Lake Biwa. In: Kawanabe, H., Nishino, M., Maehata, M. (eds). Lake Biwa: Interactions between Nature and People, pp. 491–496. SpringerGoogle Scholar
  3. 3.
    van Eerden, M.R., van Rijn, S., Volponi, S., Paquet, J.Y., Carss, D.: Cormorant and the European environment: exploring cormorant ecology on a continental scale. COST Action 635 Final Report I: 126 (2012)Google Scholar
  4. 4.
    Doucette, J.L., Wissel, B., Somers, C.M.: Cormorant-fisheries conflicts: stable isotopes reveal a consistent niche for avian piscivores in diverse food webs. Ecol. Appl. 21(8), 2987–3001 (2011)CrossRefGoogle Scholar
  5. 5.
    Takahashi, T., Kameda, K., Kawamura, M., Nakajima, T.: Food habits of great cormorant Phalacrocorax carbo hanedae at Lake Biwa, Japan, with special reference to ayu Plecoglossus altivelis altivelis. Fish. Sci. 72, 477–484 (2006)CrossRefGoogle Scholar
  6. 6.
    Yamamoto, M.: What kind of bird is Great Cormorant? (in Japanese). Accessed 30 May 2018
  7. 7.
    Ministry of Agriculture, Forestry, and Fisheries, Statistics of Fisheries in Japan during 2017. (in Japanese). Accessed 30 May 2018
  8. 8.
    Yamamoto, M.: Facing to Great Cormorant. (in Japanese). Accessed 30 May 2018
  9. 9.
    Kameda, K., Koba, K., Hobara, S., Osono, T., Terai, M.: Pattern of natural 15N abundance in lakeside forest ecosystem affected by cormorant-derived nitrogen. Hydrobiologia 567(1), 69–86 (2006)CrossRefGoogle Scholar
  10. 10.
    Pham, H.: Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, Heidelberg (2009). Scholar
  11. 11.
    Cadenillas, A.: Optimal central bank intervention in the foreign exchange market. J. Econ. Theory 87, 218–242 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ohnishi, M., Tsujimura, M.: An impulse control of a geometric Brownian motion with quadratic costs. Eur. J. Oper. Res. 168(2), 311–321 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tsujimura, M., Maeda, A.: Stochastic Control: Theory and Applications. Asakura Shoten, Tokyo (2016)Google Scholar
  14. 14.
    Federico, S., Gassiat, P.: Viscosity characterization of the value function of an investment-consumption problem in presence of an illiquid asset. J. Optim. Theory Appl. 160(3), 966–991 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Korn, R.: Portfolio optimisation with strictly positive transaction costs and impulse control. Financ. Stochast. 2(2), 85–114 (1998)CrossRefGoogle Scholar
  16. 16.
    Yaegashi, Y., Yoshioka, H., Unami, K., Fujihara, M.: Impulse and singular stochastic control approaches for management of fish-eating bird population. In: Proceedings on International Conference on Optimization and Decision Science (2018). (accepted)Google Scholar
  17. 17.
    Yaegashi, Y., Yoshioka, H., Unami, K., Fujihara, M.: Numerical computation of the coefficients in an exact solution to an impulse control model for optimal population management. In: Proceedings on Computational Engineering Conference JSCES, vol. 23 (2018). (accepted)Google Scholar
  18. 18.
    Øksendal, B.: Stochastic Differential Equations. U. Springer, Heidelberg (2003). Scholar
  19. 19.
    Yoshioka, H., Unami, K., Fujihara, M.: Mathematical analysis on a conforming finite element scheme for advection-dispersion-decay equations on connected graphs. J. JSCE. Ser. A2(70), I265–I274 (2014)Google Scholar
  20. 20.
    De Falco, C., O’Riordan, E.: A parameter robust Petrov-Galerkin scheme for advection-diffusion-reaction equations. Numer. Algorithm 56, 107–127 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Forsyth, P.A., Vetzal, K.R.: Numerical methods for nonlinear PDEs in finance. In: Duan, J.C., Härdle, W., Gentle, J. (eds.) Handbook of Computational Finance, pp. 503–528. Springer, Heidelberg (2012). Scholar
  22. 22.
    Azimzadeh, P.: Impulse control in finance: numerical methods and viscosity solutions. A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada (2017)Google Scholar
  23. 23.
    Klaij, C.M., van der Vegt, J.J., van der Ven, H.: Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 219(2), 622–643 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Witte, J.H., Reisinger, C.: A penalty method for the numerical solution of Hamilton–Jacobi–Bellman (HJB) equations in finance. SIAM J. Numer. Anal. 49(1), 213–231 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Yuta Yaegashi
    • 1
    • 2
    Email author
  • Hidekazu Yoshioka
    • 3
  • Koichi Unami
    • 1
  • Masayuki Fujihara
    • 1
  1. 1.Kyoto UniversityKyotoJapan
  2. 2.Research Fellow of Japan Society for the Promotion of ScienceTokyoJapan
  3. 3.Shimane UniversityMatsueJapan

Personalised recommendations