×

zbMATH — the first resource for mathematics

Continuous versus pulse harvesting for population models in constant and variable environment. (English) Zbl 1143.92327
Summary: We consider both autonomous and nonautonomous population models subject to either impulsive or continuous harvesting. It is demonstrated that the impulsive strategy can be as good as the continuous one, but cannot outperform it. We introduce a model, where certain harm to the population is incorporated in each harvesting event, and study it for the logistic and the Gompertz laws of growth. In this case, impulsive harvesting is not only the optimal strategy but is the only possible one.

MSC:
92D40 Ecology
34A37 Ordinary differential equations with impulses
49N90 Applications of optimal control and differential games
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Clark C.W. (1976). Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York · Zbl 0364.90002
[2] Cooke K.L. and Witten M. (1986). One-dimensional linear and logistic harvesting models. Math. Model. 7: 301–340 · Zbl 0611.92025 · doi:10.1016/0270-0255(86)90054-0
[3] Gao S.J., Chen L.S. and Sun L.H. (2005). Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos Solutions Fractals 25(5): 1209–1219 · Zbl 1065.92056 · doi:10.1016/j.chaos.2004.11.093
[4] Tang S., Cheke R.A. and Xiao Y. (2006). Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations. Nonlinear Anal. Ser. A: Theory Methods 65: 2311–2341 · Zbl 1119.39011 · doi:10.1016/j.na.2006.02.049
[5] Xu C., Boyce M.S. and Daley D.J. (2005). Harvesting in seasonal environments. J. Math. Biol. 50: 663–682 · Zbl 1066.92057 · doi:10.1007/s00285-004-0303-5
[6] Zhang X., Shuai Z. and Wang K. (2003). Optimal impulsive harvesting policy for single population. Nonlinear Anal. Real World Appl. 4: 639–651 · Zbl 1011.92052 · doi:10.1016/S1468-1218(02)00084-6
[7] Zhang Y., Xiu Z. and Chen L. (2006). Optimal impulsive harvesting of a single species with Gompertz law of growth. J. Biol. Syst. 14: 303–314 · Zbl 1109.92062 · doi:10.1142/S0218339006001829
[8] Fan M. and Wang K. (1998). Optimal harvesting policy for single population with periodic coefficients. Math. Biosci. 152: 165–177 · Zbl 0940.92030 · doi:10.1016/S0025-5564(98)10024-X
[9] Xiao Y., Cheng D. and Qin H. (2006). Optimal impulsive control in periodic ecosystems. Syst. Control Lett. 55: 556–565 · Zbl 1129.49308
[10] Angelova J. and Dishliev A. (2000). Optimization problems for one-impulsive models from population Dynamics. Nonlinear Anal. 39: 483–497 · Zbl 0942.34010 · doi:10.1016/S0362-546X(98)00216-8
[11] Braverman E., Israeli M., Averbuch A. and Vozovoi L. (1998). A fast 3-D Poisson solver of arbitrary order accuracy. J. Comput. Phys. 144: 109–136 · Zbl 0910.35027 · doi:10.1006/jcph.1998.6001
[12] Boyce M.S. and Daley D.J. (1980). Population tracking of fluctuating environments and natural selection for tracking ability. Am. Nat. 115: 480–491 · doi:10.1086/283575
[13] Chen L., Sun L. and Dong (2007). Optimal harvesting policies for periodic Gompertz systems. Nonlinear Anal. Real World Appl. 8: 572–578 · Zbl 1152.34333 · doi:10.1016/j.nonrwa.2006.01.001
[14] Berezansky L. and Braverman E. (2004). On impulsive Beverton–Holt difference equations and their applications. J. Differ. Equ. Appl. 10: 851–868 · Zbl 1068.39005 · doi:10.1080/10236190410001726421
[15] Braverman E. (2005). On a discrete model of population dynamics with impulsive harvesting or recruitment. Nonlinear Anal. Ser. A: Theory Methods 63: e751–e759 · Zbl 1222.92060 · doi:10.1016/j.na.2004.12.015
[16] Braverman E. and Kinzebulatov D. (2006). On linear perturbations of the Ricker model. Math. Biosci. 202: 323–339 · Zbl 1097.92052 · doi:10.1016/j.mbs.2006.04.008
[17] Schreiber A.J. (2001). Chaos and population disappearances in simple ecological models. J. Math. Biol. 42: 239–260 · Zbl 0977.92032 · doi:10.1007/s002850000070
[18] Braverman, E., Mamdani, R.: On optimal impulsive sustainable harvesting, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14(S2) (suppl.), Adv Dynamical Syst., 112–116 (2007)
[19] Artstein Z. (1993). Chattering limit for a model of harvesting in a rapidly changing environment. Appl. Math. Optim. 28: 133–147 · Zbl 0795.90006 · doi:10.1007/BF01182977
[20] Ludwig D. (1980). Harvesting strategies for a randomly fluctuating population. J. Cons. Int. Explor. Mer. 39: 168–174
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.