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Optimal States of Distributed Exploited Populations with Periodic Impulse Harvesting

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Abstract

The dynamics of a population distributed on a torus is described by an equation of the Kolmogorov–Petrovsky–Piskunov–Fisher type in the divergence form. The population is exploited by periodic harvesting of a constant distributed measurable fraction of its density. We prove that there exists a harvesting ratio maximizing the time-averaged income in kind, i.e., a ratio that provides an optimal stationary exploitation in the long run.

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REFERENCES

  1. S. M. Aseev, K. O. Besov, and S. Yu. Kaniovski, “Optimal policies in the Dasgupta–Heal–Solow–Stiglitz model under nonconstant returns to scale,” Proc. Steklov Inst. Math. 304, 74–109 (2019). 10.1134/S0081543819010061 https://doi.org/10.1134/S0081543819010061

    Article  MathSciNet  MATH  Google Scholar 

  2. S. M. Aseev and V. M. Veliov, “Another view of the maximum principle for infinite-horizon optimal control problems in economics,” Russ. Math. Surv. 74 (6), 963–1011 (2019). https://doi.org/10.1070/RM9915

    Article  MathSciNet  MATH  Google Scholar 

  3. A. O. Belyakov and A. A. Davydov, “Efficiency optimization for the cyclic use of a renewable resource,” Proc. Steklov Inst. Math. 299 (Suppl. 1), S14–S21 (2017). https://doi.org/10.1134/S0081543817090036

    Article  MATH  Google Scholar 

  4. H. Berestycki, F. Hamel, and L. Roques, “Analysis of the periodically fragmented environment model: I — Species persistence,” J. Math. Biol. 51 (1), 75–113. (2005). https://doi.org/10.1007/s00285-004-0313-3

    Article  MathSciNet  MATH  Google Scholar 

  5. A. O. Belyakov, A. A. Davydov, and V. M. Veliov, “Optimal cyclic exploitation of renewable resources,” J. Dyn. Control Syst. 21 (3), 475–494 (2015). https://doi.org/10.1007/s10883-015-9271-x

    Article  MathSciNet  MATH  Google Scholar 

  6. A. O. Belyakov, A. A. Davydov, and V. M. Veliov, “Optimal cyclic harvesting of renewable resource,” Dokl. Math. 96 (2), 472–474 (2017). https://doi.org/10.1134/S1064562417050180

    Article  MathSciNet  MATH  Google Scholar 

  7. P. J. Cohen and S. J. Foale, “Sustaining small-scale fisheries with periodically harvested marine reserves,” Mar. Policy 37, 278–287 (2013). https://doi.org/10.1016/j.marpol.2012.05.010

    Article  Google Scholar 

  8. D. Daners and P. Medina, Abstract Evolution Equations, Periodic Problems and Applications (Longman Sci. Tech., London, 1992), Ser. Pitman Research Notes in Mathematics, Vol. 279.

    MATH  Google Scholar 

  9. A. A. Davydov, “Existence of optimal stationary states of exploited populations with diffusion,” Proc. Steklov Inst. Math. 310, 124–130 (2020). https://doi.org/10.1134/S0081543820050090

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Davydov, “Optimal steady state of distributed population in periodic environment,” AIP Conf. Proc. 2333, article 120007 (2021). https://doi.org/10.1063/5.0041960

    Article  Google Scholar 

  11. Y. Du and R. Peng, “The periodic logistic equation with spatial and temporal degeneracies,” Trans. Am. Math. Soc. 364 (11), 6039–6070 (2012). https://doi.org/10.1090/S0002-9947-2012-05590-5

    Article  MathSciNet  MATH  Google Scholar 

  12. R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7 (4), 355–369 (1937). 10.1111/j.1469-1809.1937.tb02153.x https://doi.org/10.1111/j.1469-1809.1937.tb02153.x

    Article  MATH  Google Scholar 

  13. K. Henderson and M. Loreau, “An ecological theory of changing human population dynamics,” People Nat. 1 (1), 31–43 (2019). https://doi.org/10.1002/pan3.8

    Article  Google Scholar 

  14. P. Hess, Periodic–Parabolic Boundary Value Problems and Positivity, (Wiley, New York, 1991), Ser. Pitman Research Notes in Mathematics, Vol. 247.

    MATH  Google Scholar 

  15. A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Moscow Univ. Math. Mech. 1 (6), 333–358 (1937); reprinted in Selected Works of A. N. Kolmogorov, Vol. I, Ed. by V. M. Tikhomirov (Kluwer, Dordrecht, 1991), pp. 242–270.

    Google Scholar 

  16. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1968).

    MATH  Google Scholar 

  17. P. K. Medina, “Feedback stabilizability of time-periodic parabolic equations,” in Dynamics Reported: Expositions in Dynamical Systems, Ed. by C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther (Springer, Berlin, 1996), Vol. 5, pp. 26–98.

    Chapter  Google Scholar 

  18. G. Nadin, “Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation,” J. Diff. Equations 249 (6), 1288–1304 (2010). https://doi.org/10.1016/j.jde.2010.05.007

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Undersander, B. Albert, D. Cosgrove, D. Johnson, and P. Peterson, Pastures for Profit: A Guide to Rotational Grazing (Cooperative Extension Publ., Madison, WI, 2002).

    Google Scholar 

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Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation (first author, results on the properties of the quality functional, project no. 0718-2020-0025) and by the Russian Science Foundation (both authors, the main theorem, project no. 19-11-00223).

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Authors and Affiliations

  1. Moscow State University, 119991, Moscow, Russia

    A. A. Davydov & D. A. Melnik

  2. National University of Science and Technology MISIS, 119049, Moscow, Russia

    A. A. Davydov

Authors
  1. A. A. Davydov
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  2. D. A. Melnik
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Corresponding authors

Correspondence to A. A. Davydov or D. A. Melnik.

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The authors declare that they have no conflicts of interest.

Additional information

Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 27, No. 2, pp. 99 - 107, 2021 https://doi.org/10.21538/0134-4889-2021-27-2-99-107.

Translated by I. Tselishcheva

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Davydov, A.A., Melnik, D.A. Optimal States of Distributed Exploited Populations with Periodic Impulse Harvesting. Proc. Steklov Inst. Math. 315 (Suppl 1), S81–S88 (2021). https://doi.org/10.1134/S0081543821060079

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