Abstract
This paper deals with designing a harvesting control strategy for a predator–prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baek H (2010) Dynamic analysis of an impulsively controlled predator–prey system. Electron J Qual Theory Differ Equ 2010(19):1–14
Beddington JR, Cooke JG (1982) Harvesting from a prey–predator complex. Ecol Model 14:155–177
Beddington JR, May RM (1980) Maximum sustainable yields in systems subject to harvesting at more than one trophic level. Math Biosci 51:261–281
Brauer F, Soudack AC (1979a) Stability regions and transition phenomena for harvested predator–prey systems. Math Biol 7:319–337
Brauer F, Soudack AC (1979b) Stability regions in predator–prey systems with constant-rate prey harvesting. Math Biol 8:55–71
Brauer F, Soudack AC (1981) Coexistence properties of some predator–prey systems under constant rate harvesting and stocking. Math Biol 12:101–114
Chen J, Huang J, Ruan S, Wang J (2013) Bifurcation of invariant tori in predator–prey models with seasonal prey harvesting. SIAM J Appl Math 73:1876–1905
Cockburn JC, McLoud K, Wagner J (2011) Robust control of predator–prey–hunter systems. In: American control conference (ACC). IEEE, pp 1934–1939
Dai G, Tang M (1998) Coexistence region and global dynamics of a harvested predator–prey system. SIAM J Appl Math 58:193–210
Dyar JA, Wagner J (2003) Uncertainty and species recovery program design. J Environ Econ Manag 45(2):505–522
El-Gohary A, Al-Ruzaiza A (2007) Chaos and adaptive control in two prey, one predator system with nonlinear feedback. Chaos Solitons Fractals 34(2):443–453
Fischer N, Kamalapurkar R, Dixon WE (2013) Lasalle-Yoshizawa corollaries for nonsmooth systems. IEEE Trans Autom Control 9(58):2333–2338
Hoekstra J, van den Bergh JC (2005) Harvesting and conservation in a predator–prey system. J Econ Dyn Control 29(6):1097–1120
Huang J, Chen J, Gong Y, Zhang W (2013a) Complex dynamics in predator–prey models with nonmonotonic functional response and harvesting. Math Model Nat Phenom 8(5):95–118
Huang J, Gong Y, Chen J (2013b) Multiple bifurcations in a predator–prey system of holling and leslie type with constant-yield prey harvesting. Int J Bifurc Chaos 23(10):1350,164
Huang J, Gong Y, Ruan S (2013c) Bifurcation analysis in a predator–prey model with constant-yield predator harvesting. DCDS-B 18:2101–2121
Huang J, Liu S, Ruan S, Zhang X (2016) Bogdanov-takens bifurcation of codimension 3 in a predator–prey model with constant-yield predator harvesting. Commun Pure Appl Anal 15(3):1041–1055
Krstic M, Kokotovic PV, Kanellakopoulos I (1995) Nonlinear and adaptive control design. Wiley, New York
McCarthy MA, Armstrong DP, Runge MC (2012) Adaptive management of reintroduction. Reintrod Biol Integr Sci Manag 12:256
Rao VLGM, Rasappan S, Murugesan R, Srinivasan V (2015) A prey predator model with vulnerable infected prey consisting of non-linear feedback. Appl Math Sci 9(42):2091–2102
Rondeau D (2001) Along the way back from the brink. J Environ Econ Manag 42(2):156–182
Sastry S (2013) Nonlinear systems: analysis, stability, and control, vol 10. Springer, New York
Sen M, Srinivasu PDN, Banerjee M (2015) Global dynamics of an additional food provided predator–prey system with constant harvest in predators. Appl Math Comput 250:193–211
Utkin VI (2013) Sliding modes in control and optimization. Springer, New York
Wang L, Xu R, Feng G (2011) Analysis of a stage-structured predator–prey system concerning impulsive control strategy. Differ Equ Dyn Syst 19(4):303–313
Xia J, Liu Z, Yuan R, Ruan S (2009) The effects of harvesting and time delay on predator–prey systems with holling type II functional response. SIAM J Appl Math 70:1178–1200
Xiao D, Jennings LS (2005) Bifurcations of a ratio-dependent predator–prey with constant rate harvesting. SIAM J Appl Math 65:737–753
Zhao C, Wang M, Zhao P (2005) Optimal control of harvesting for age-dependent predator–prey system. Math Comput Model 42(5):573–584
Zhu CR, Lan KQ (2010) Phase portraits, Hopf bifurcation and limit cycles of Leslie–Gower predator–prey systems with harvesting rates. DCDS-B 14:289–306
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sen, M., Simha, A. & Raha, S. Adaptive Control Based Harvesting Strategy for a Predator–Prey Dynamical System. Acta Biotheor 66, 293–313 (2018). https://doi.org/10.1007/s10441-018-9323-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10441-018-9323-1