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2016, 13(4): 673-695. doi: 10.3934/mbe.2016014

Optimal harvesting policy for the Beverton--Holt model

Martin Bohner 1, and  Sabrina Streipert 1,

1. 

Missouri University of Science and Technology, 400 West, 12th Street, Rolla, MO 65409-0020, United States, United States

Received  September 2015 Revised  December 2015 Published  May 2016

In this paper, we establish the exploitation of a single population modeled by the Beverton--Holt difference equation with periodic coefficients. We begin our investigation with the harvesting of a single autonomous population with logistic growth and show that the harvested logistic equation with periodic coefficients has a unique positive periodic solution which globally attracts all its solutions. Further, we approach the investigation of the optimal harvesting policy that maximizes the annual sustainable yield in a novel and powerful way; it serves as a foundation for the analysis of the exploitation of the discrete population model. In the second part of the paper, we formulate the harvested Beverton--Holt model and derive the unique periodic solution, which globally attracts all its solutions. We continue our investigation by optimizing the sustainable yield with respect to the harvest effort. The results differ from the optimal harvesting policy for the continuous logistic model, which suggests a separate strategy for populations modeled by the Beverton--Holt difference equation.
Keywords: maximum sustainable yield., Difference equations, logistic growth, periodic solution, weighted Jensen inequality, global stability.
Mathematics Subject Classification: Primary: 39A23, 92D25; Secondary: 39A30, 49K1.
Citation: Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014
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show all references

References:
[1]

Appl. Math. Comput., 187 (2007), 873-882. doi: 10.1016/j.amc.2006.09.007.  Google Scholar

[2]

in Monographs in Inequalities, ELEMENT, Zagreb, Volume 9, 2015. Google Scholar

[3]

J. Differ. Equations Appl., 10 (2004), 851-868. doi: 10.1080/10236190410001726421.  Google Scholar

[4]

Fish & Fisheries Series, 1993. doi: 10.1007/978-94-011-2106-4.  Google Scholar

[5]

Math. Biosci., 135 (1996), 111-127. doi: 10.1016/0025-5564(95)00170-0.  Google Scholar

[6]

J. Biol. Dyn., 7 (2013), 86-95. Google Scholar

[7]

Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[8]

Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[9]

in Discrete dynamics and difference equations, World Sci. Publ., Hackensack, NJ, (2010), 189-193. doi: 10.1142/9789814287654_0012.  Google Scholar

[10]

Int. J. Math. Comput., 26 (2015), 1-10.  Google Scholar

[11]

in Difference equations, discrete dynamical systems, and applications, Springer-Verlag, Berlin-Heidelberg-New York, 150 (2015), 3-14. doi: 10.1007/978-3-319-24747-2_1.  Google Scholar

[12]

Appl. Anal., 86 (2007), 1007-1015. doi: 10.1080/00036810701474140.  Google Scholar

[13]

Nonlinear Anal., 71 (2009), e2173-e2181. doi: 10.1016/j.na.2009.04.025.  Google Scholar

[14]

J. Math. Biol., 57 (2008), 413-434. doi: 10.1007/s00285-008-0169-z.  Google Scholar

[15]

John Wiley & Sons, Inc., New York, 1990.  Google Scholar

[16]

Appl. Math. Lett., 36 (2014), 19-24. doi: 10.1016/j.aml.2014.04.011.  Google Scholar

[17]

Math. Biosci., 152 (1998), 165-177. doi: 10.1016/S0025-5564(98)10024-X.  Google Scholar

[18]

Volume 8, Elsevier North-Holland, Inc., New York, NY, 1980. Google Scholar

[19]

Fisheries Research, 24 (1995), 3-8. doi: 10.1016/0165-7836(95)00377-M.  Google Scholar

[20]

Academic Press, Inc., Boston, MA, 1991.  Google Scholar

[21]

Int. J. Difference Equ., 7 (2012), 35-60.  Google Scholar

[22]

J. Difference Equ. Appl., 11 (2005), 415-422. doi: 10.1080/10236190412331335463.  Google Scholar

[23]

J. Difference Equ. Appl., 20 (2014), 859-874. doi: 10.1080/10236198.2013.824968.  Google Scholar

[24]

Appl. Math. Optim., 31 (1995), 219-241. doi: 10.1007/BF01182789.  Google Scholar

[25]

Int. J. Difference Equ., 4 (2009), 115-136.  Google Scholar

[26]

J. Math. Inequal., 5 (2011), 253-264. doi: 10.7153/jmi-05-23.  Google Scholar

[27]

Mar. Resour. Econ., 24 (2009), 147-169. Google Scholar

[28]

(Russian) (Chita) in Modeling of natural systems and optimal control problems, VO "Nauka'', Novosibirsk, (1993), 65-74.  Google Scholar

[29]

Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42. Google Scholar

[30]

Adv. Dyn. Syst. Appl., 1 (2006), 113-120.  Google Scholar

[31]

J. Math. Biol., 50 (2005), 663-682. doi: 10.1007/s00285-004-0303-5.  Google Scholar

[32]

Nonlinear Anal. Real World Appl., 4 (2003), 639-651. doi: 10.1016/S1468-1218(02)00084-6.  Google Scholar

[33]

IEEE Trans. Automat. Cont., 40 (1995), 1779-1783. doi: 10.1109/9.467682.  Google Scholar

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