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

## Optimal harvesting policy for the Beverton--Holt model

 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.
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
##### References:
 [1] L. Bai and K. Wang, A diffusive single-species model with periodic coefficients and its optimal harvesting policy, Appl. Math. Comput., 187 (2007), 873-882. doi: 10.1016/j.amc.2006.09.007.  Google Scholar [2] J. Barić, R. Bibi, M. Bohner, A. Nosheen and J. Pečarić, Jensen inequalities on time scales, in Monographs in Inequalities, ELEMENT, Zagreb, Volume 9, 2015. Google Scholar [3] L. Berezansky and E. Braverman, On impulsive Beverton-Holt difference equations and their applications, J. Differ. Equations Appl., 10 (2004), 851-868. doi: 10.1080/10236190410001726421.  Google Scholar [4] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Fish & Fisheries Series, 1993. doi: 10.1007/978-94-011-2106-4.  Google Scholar [5] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two-species system. I, Math. Biosci., 135 (1996), 111-127. doi: 10.1016/0025-5564(95)00170-0.  Google Scholar [6] M. Bohner and R. Chieochan, The Beverton-Holt $q$-difference equation, J. Biol. Dyn., 7 (2013), 86-95. Google Scholar [7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar [8] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar [9] M. Bohner, S. Stević and H. Warth, The Beverton-Holt difference equation, in Discrete dynamics and difference equations, World Sci. Publ., Hackensack, NJ, (2010), 189-193. doi: 10.1142/9789814287654_0012.  Google Scholar [10] M. Bohner and S. H. Streipert, The Beverton-Holt equation with periodic growth rate, Int. J. Math. Comput., 26 (2015), 1-10.  Google Scholar [11] M. Bohner and S. H. Streipert, The Beverton-Holt $q$-difference equation with periodic growth rate, 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] M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015. doi: 10.1080/00036810701474140.  Google Scholar [13] E. Braverman and L. Braverman, Optimal harvesting of diffusive models in a nonhomogeneous environment, Nonlinear Anal., 71 (2009), e2173-e2181. doi: 10.1016/j.na.2009.04.025.  Google Scholar [14] E. Braverman and R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol., 57 (2008), 413-434. doi: 10.1007/s00285-008-0169-z.  Google Scholar [15] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley & Sons, Inc., New York, 1990.  Google Scholar [16] T. Diagana, Almost automorphic solutions to a Beverton-Holt dynamic equation with survival rate, Appl. Math. Lett., 36 (2014), 19-24. doi: 10.1016/j.aml.2014.04.011.  Google Scholar [17] M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci., 152 (1998), 165-177. doi: 10.1016/S0025-5564(98)10024-X.  Google Scholar [18] B.-S. Goh, Management and Analysis of Biological Populations, Volume 8, Elsevier North-Holland, Inc., New York, NY, 1980. Google Scholar [19] M. Holden, Beverton and Holt revisited, Fisheries Research, 24 (1995), 3-8. doi: 10.1016/0165-7836(95)00377-M.  Google Scholar [20] W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Inc., Boston, MA, 1991.  Google Scholar [21] C. Kent, V. Kocic and Y. Kostrov, Attenuance and resonance in a periodically forced sigmoid Beverton-Holt model, Int. J. Difference Equ., 7 (2012), 35-60.  Google Scholar [22] V. L. Kocic, A note on the nonautonomous Beverton-Holt model, J. Difference Equ. Appl., 11 (2005), 415-422. doi: 10.1080/10236190412331335463.  Google Scholar [23] V. L. Kocic and Y. Kostrov, Dynamics of a discontinuous discrete Beverton-Holt model, J. Difference Equ. Appl., 20 (2014), 859-874. doi: 10.1080/10236198.2013.824968.  Google Scholar [24] A. W. Leung, Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim., 31 (1995), 219-241. doi: 10.1007/BF01182789.  Google Scholar [25] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, Two modifications of the Beverton-Holt equation, Int. J. Difference Equ., 4 (2009), 115-136.  Google Scholar [26] Z. Pavić, J. Pečarić and I. Perić, Integral, discrete and functional variants of Jensen's inequality, J. Math. Inequal., 5 (2011), 253-264. doi: 10.7153/jmi-05-23.  Google Scholar [27] O. Tahvonen, Optimal harvesting of age-structured fish populations, Mar. Resour. Econ., 24 (2009), 147-169. Google Scholar [28] G. V. Tsvetkova, Construction of an optimal policy taking into account ecological constraints, (Russian) (Chita) in Modeling of natural systems and optimal control problems, VO "Nauka'', Novosibirsk, (1993), 65-74.  Google Scholar [29] P.-F. Verhulst, Recherches mathématiques sur la loi d'accroissement de la population, Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42. Google Scholar [30] F.-H. Wong, C.-C. Yeh and W.-C. Lian, An extension of Jensen's inequality on time scales, Adv. Dyn. Syst. Appl., 1 (2006), 113-120.  Google Scholar [31] C. Xu, M. S. Boyce and J. D. Daley, Harvesting in seasonal environments, J. Math. Biol., 50 (2005), 663-682. doi: 10.1007/s00285-004-0303-5.  Google Scholar [32] X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. Real World Appl., 4 (2003), 639-651. doi: 10.1016/S1468-1218(02)00084-6.  Google Scholar [33] M. Ziolko and J. Kozlowski, Some optimization models of growth in biology, IEEE Trans. Automat. Cont., 40 (1995), 1779-1783. doi: 10.1109/9.467682.  Google Scholar

show all references

##### References:
 [1] L. Bai and K. Wang, A diffusive single-species model with periodic coefficients and its optimal harvesting policy, Appl. Math. Comput., 187 (2007), 873-882. doi: 10.1016/j.amc.2006.09.007.  Google Scholar [2] J. Barić, R. Bibi, M. Bohner, A. Nosheen and J. Pečarić, Jensen inequalities on time scales, in Monographs in Inequalities, ELEMENT, Zagreb, Volume 9, 2015. Google Scholar [3] L. Berezansky and E. Braverman, On impulsive Beverton-Holt difference equations and their applications, J. Differ. Equations Appl., 10 (2004), 851-868. doi: 10.1080/10236190410001726421.  Google Scholar [4] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Fish & Fisheries Series, 1993. doi: 10.1007/978-94-011-2106-4.  Google Scholar [5] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two-species system. I, Math. Biosci., 135 (1996), 111-127. doi: 10.1016/0025-5564(95)00170-0.  Google Scholar [6] M. Bohner and R. Chieochan, The Beverton-Holt $q$-difference equation, J. Biol. Dyn., 7 (2013), 86-95. Google Scholar [7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar [8] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar [9] M. Bohner, S. Stević and H. Warth, The Beverton-Holt difference equation, in Discrete dynamics and difference equations, World Sci. Publ., Hackensack, NJ, (2010), 189-193. doi: 10.1142/9789814287654_0012.  Google Scholar [10] M. Bohner and S. H. Streipert, The Beverton-Holt equation with periodic growth rate, Int. J. Math. Comput., 26 (2015), 1-10.  Google Scholar [11] M. Bohner and S. H. Streipert, The Beverton-Holt $q$-difference equation with periodic growth rate, 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] M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015. doi: 10.1080/00036810701474140.  Google Scholar [13] E. Braverman and L. Braverman, Optimal harvesting of diffusive models in a nonhomogeneous environment, Nonlinear Anal., 71 (2009), e2173-e2181. doi: 10.1016/j.na.2009.04.025.  Google Scholar [14] E. Braverman and R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol., 57 (2008), 413-434. doi: 10.1007/s00285-008-0169-z.  Google Scholar [15] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley & Sons, Inc., New York, 1990.  Google Scholar [16] T. Diagana, Almost automorphic solutions to a Beverton-Holt dynamic equation with survival rate, Appl. Math. Lett., 36 (2014), 19-24. doi: 10.1016/j.aml.2014.04.011.  Google Scholar [17] M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci., 152 (1998), 165-177. doi: 10.1016/S0025-5564(98)10024-X.  Google Scholar [18] B.-S. Goh, Management and Analysis of Biological Populations, Volume 8, Elsevier North-Holland, Inc., New York, NY, 1980. Google Scholar [19] M. Holden, Beverton and Holt revisited, Fisheries Research, 24 (1995), 3-8. doi: 10.1016/0165-7836(95)00377-M.  Google Scholar [20] W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Inc., Boston, MA, 1991.  Google Scholar [21] C. Kent, V. Kocic and Y. Kostrov, Attenuance and resonance in a periodically forced sigmoid Beverton-Holt model, Int. J. Difference Equ., 7 (2012), 35-60.  Google Scholar [22] V. L. Kocic, A note on the nonautonomous Beverton-Holt model, J. Difference Equ. Appl., 11 (2005), 415-422. doi: 10.1080/10236190412331335463.  Google Scholar [23] V. L. Kocic and Y. Kostrov, Dynamics of a discontinuous discrete Beverton-Holt model, J. Difference Equ. Appl., 20 (2014), 859-874. doi: 10.1080/10236198.2013.824968.  Google Scholar [24] A. W. Leung, Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim., 31 (1995), 219-241. doi: 10.1007/BF01182789.  Google Scholar [25] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, Two modifications of the Beverton-Holt equation, Int. J. Difference Equ., 4 (2009), 115-136.  Google Scholar [26] Z. Pavić, J. Pečarić and I. Perić, Integral, discrete and functional variants of Jensen's inequality, J. Math. Inequal., 5 (2011), 253-264. doi: 10.7153/jmi-05-23.  Google Scholar [27] O. Tahvonen, Optimal harvesting of age-structured fish populations, Mar. Resour. Econ., 24 (2009), 147-169. Google Scholar [28] G. V. Tsvetkova, Construction of an optimal policy taking into account ecological constraints, (Russian) (Chita) in Modeling of natural systems and optimal control problems, VO "Nauka'', Novosibirsk, (1993), 65-74.  Google Scholar [29] P.-F. Verhulst, Recherches mathématiques sur la loi d'accroissement de la population, Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42. Google Scholar [30] F.-H. Wong, C.-C. Yeh and W.-C. Lian, An extension of Jensen's inequality on time scales, Adv. Dyn. Syst. Appl., 1 (2006), 113-120.  Google Scholar [31] C. Xu, M. S. Boyce and J. D. Daley, Harvesting in seasonal environments, J. Math. Biol., 50 (2005), 663-682. doi: 10.1007/s00285-004-0303-5.  Google Scholar [32] X. Zhang, Z. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. Real World Appl., 4 (2003), 639-651. doi: 10.1016/S1468-1218(02)00084-6.  Google Scholar [33] M. Ziolko and J. Kozlowski, Some optimization models of growth in biology, IEEE Trans. Automat. Cont., 40 (1995), 1779-1783. doi: 10.1109/9.467682.  Google Scholar
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