Advertisement

Optimal Spatial Harvesting Strategy and Symmetry-Breaking

  • Kazuhiro KurataEmail author
  • Junping Shi
Article

Abstract

A reaction-diffusion model with logistic growth and constant effort harvesting is considered. By minimizing an intrinsic biological energy function, we obtain an optimal spatial harvesting strategy which will benefit the population the most. The symmetry properties of the optimal strategy are also discussed, and related symmetry preserving and symmetry breaking phenomena are shown with several typical examples of habitats.

Keywords

Reaction-diffusion Harvesting Optimal spatial effort function Symmetry breaking 

References

  1. 1.
    Alikakos, N., Bates, P.W., Chen, X.: Convergence of the Cahn-Hillard equation to Hele-Shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Block, B.A., Dewar, H., Blackwell, S.B., Williams, T.D., Prince, E.D., Farwell, C.J., Boustany, A., Teo, S.L.H., Seitz, A., Walli, A., Fudge, D.: Migratory movements, depth preferences, and thermal biology of Atlantic bluefin tuna. Science 293, 1310–1314 (2001) CrossRefGoogle Scholar
  3. 3.
    Bates, P.W., Fife, P.C.: The dynamics of nucleating for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53(4), 990–1008 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brauer, F., Castillo-Chávez, C.: Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics, vol. 40. Springer, New York (2001) zbMATHGoogle Scholar
  5. 5.
    Cantrell, R.S., Cosner, C.: Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. R. Soc. Edinb. Sect. A 112(3–4), 293–318 (1989) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equation. Wiley Series in Mathematical and Computational Biology. Wiley, New York (2003) Google Scholar
  7. 7.
    Carr, J., Gurtin, M.E., Slemrod, M.: Structured phase transitions on a finite interval. Arch. Ration. Mech. Anal. 86(4), 317–351 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chanillo, S., Grieser, M., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214, 315–337 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chanillo, S., Grieser, M., Kurata, K.: The free boundary problem in the optimization of composite membranes. Contemp. Math. 268, 61–81 (2000) MathSciNetGoogle Scholar
  10. 10.
    Clark, C.W.: Mathematical Bioeconomics, The Optimal Management of Renewable Resources. Wiley, New York (1991) Google Scholar
  11. 11.
    Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229(1), 63–91 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fraenkel, L.E.: Introduction to Maximum Principles and Symmetry in Elliptic Equations. Cambridge University Press, Cambridge (2000) Google Scholar
  13. 13.
    Gurtin, M.E., Matano, H.: On the structure of equilibrium phase transitions with the gradient theory of fluids. Q. Appl. Math. 46, 301–317 (1988) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Harrell, E.M., Kröger, P., Kurata, K.: On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal. 33, 240–259 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kurata, K., Shibata, M., Sakamoto, S.: Symmetry-breaking phenomena in an optimization problem for some nonlinear elliptic equation. Appl. Math. Optim. 50, 259–278 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lieb, E., Loss, M.: Analysis. American Mathematical Society, Providence (1997) Google Scholar
  17. 17.
    Lou, Y., Yanagida, E.: Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. Jpn. J. Ind. Appl. Math. 23(3), 275–292 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Murray, J.D.: Mathematical Biology. I. An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002) Google Scholar
  19. 19.
    Murray, J.D.: Mathematical Biology. II. Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, vol. 18. Springer, New York (2003) Google Scholar
  20. 20.
    Neubert, M.G.: Marine reserves and optimal harvesting. Ecol. Lett. 6, 843–849 (2003) CrossRefGoogle Scholar
  21. 21.
    Ni, W.-M., Wei, J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Commun. Pure Appl. Math. 48(7), 731–768 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Okubo, A., Levin, S.: Diffusion and Ecological Problems: Modern Perspectives, 2nd edn. Interdisciplinary Applied Mathematics, vol. 14. Springer, New York (2001) Google Scholar
  23. 23.
    Oruganti, S., Shi, J., Shivaji, R.: Diffusive logistic equation with constant yield harvesting. I: Steady states. Trans. Am. Math. Soc. 354(9), 3601–3619 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Shi, J., Shivaji, R.: Global bifurcation for concave semiposition problems. In: Goldstein, G.R., Nagel, R., Romanelli, S. (eds.) Advances in Evolution Equations. Proceedings in Honor of J.A. Goldstein’s 60th Birthday, pp. 385–398. Dekker, New York (2003) Google Scholar
  25. 25.
    Shi, J., Shivaji, R.: Persistence in reaction diffusion models with weak Allee effect. J. Math. Biol. 52(6), 807–829 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Metropolitan UniversityHachioji-shiJapan
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA
  3. 3.School of MathematicsHarbin Normal UniversityHarbinChina

Personalised recommendations