##
**A control theory point of view on Beverton-Holt equation in population dynamics and some of its generalizations.**
*(English)*
Zbl 1137.92034

Summary: This paper is devoted to develop some “ad hoc” Control Theory formalism useful for the famous Beverton-Holt equation arising in population dynamics. In particular, the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate. For that purpose, the environment carrying capacity gains are allowed to be time-varying and designed for control purposes. The controllability property is also investigated on this dynamic extended system as well as the stability, equilibrium points and attractor oscillating trajectories. The properties of the dynamic system associated with the Beverton-Holt inverse equation allow to extrapolate in a simple dual way the above properties to the standard Beverton-Holt equation. Some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances. In all cases, a reference model being also of Beverton-Holt type is proposed to be followed by the control system.

### MSC:

92D40 | Ecology |

39A11 | Stability of difference equations (MSC2000) |

37N35 | Dynamical systems in control |

37N25 | Dynamical systems in biology |

PDFBibTeX
XMLCite

\textit{M. de la Sen} and \textit{S. Alonso-Quesada}, Appl. Math. Comput. 199, No. 2, 464--481 (2008; Zbl 1137.92034)

Full Text:
DOI

### References:

[1] | Barrowman, N. J.; Myers, R. A.; Hilborn, R.; Kehler, D. G.; Field, C. A., The variability among populations of coho salmon in the maximum productive rate and depensation, Ecological Applications, 13, 3, 784-793 (2003) |

[2] | Jensen, A. L., Harvest reference points for the Beverton and Holt dynamic pool model, Fisheries Research, 47, 93-96 (2000) |

[3] | Holden, M., Beverton and Holt revisited, Fisheries Research, 24, 3-8 (1995) |

[4] | G. Stefansson, Fish 480 (stockrec) Spawning stock, Recruitment and Production, Course at the Department of Biology of the University of Iceland using data of the Marine Research Institute of Reykjavik, Iceland, November 2005.; G. Stefansson, Fish 480 (stockrec) Spawning stock, Recruitment and Production, Course at the Department of Biology of the University of Iceland using data of the Marine Research Institute of Reykjavik, Iceland, November 2005. |

[5] | Beverton, R. J.H.; Holt, S. J., On the dynamics of exploited fish populations, Fisheries Investigation, 1 (1957) |

[6] | Kocic, V. L., A note on the nonautonomous Beverton-Holt model, Journal of Difference Equations and Applications, 11, 4-5, 415-422 (2005) · Zbl 1084.39007 |

[7] | Cushing, J. M.; Henson, S. M., A periodically forced Beverton-Holt equation, Journal of Difference Equations and Applications, 8, 12, 1119-1120 (2002) · Zbl 1023.39013 |

[8] | Cushing, J. M.; Henson, S. M., Global dynamics of some periodically forced monotone difference equations, Journal of Difference Equations and Applications, 7, 12, 859-872 (2001) · Zbl 1002.39003 |

[9] | Elaydi, S.; Sacker, R. J., Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, Journal of Difference Equations and Applications, 11, 4-5, 337-346 (2005) · Zbl 1084.39005 |

[10] | Elaydi, S.; Sacker, R. J.; equations, Periodic difference, population biology and the Cushing-Henson conjectures, Mathematical Biosciences, 201, 1-2, 195-207 (2006) · Zbl 1105.39006 |

[11] | S. Stevic, A short proof of the Cushing-Henson conjecture, Discrete Dynamics in Nature and Society vol. 2006, Article ID 37264, 5 pages, 2006, doi:10.1155/DDNS/2006/37264; S. Stevic, A short proof of the Cushing-Henson conjecture, Discrete Dynamics in Nature and Society vol. 2006, Article ID 37264, 5 pages, 2006, doi:10.1155/DDNS/2006/37264 · Zbl 1149.39300 |

[12] | Berezansky, L.; Braverman, E., On impulsive Beverton-Holt difference equations and their applications, Journal of Difference Equations and Applications, 10, 9, 851-868 (2004) · Zbl 1068.39005 |

[13] | E.M. Elabbasy, H.A. El-Metwally, E.M. Elsayed, On the difference equation \(XN AX N BX N CX N DX N\) doi:10.1155/ade/2006/82579; E.M. Elabbasy, H.A. El-Metwally, E.M. Elsayed, On the difference equation \(XN AX N BX N CX N DX N\) doi:10.1155/ade/2006/82579 · Zbl 1139.39304 |

[14] | De la Sen, M.; Alonso-Quesada, S., Model matching via multirate sampling with fast sampled input guaranteeing the stability of the plant zeros: extensions to adaptive control, IEE Control Theory and Applications, 1, 1, 210-225 (2007) |

[15] | McCarthy, M. A., The Allee effect, finding mates and theoretical models, Ecological Modelling, 103, 99-102 (1997) |

[16] | Hui, Cang, Carrying capacity, population equilibrium, and environment’s maximal load, Ecological Modelling, 192, 317-320 (2006) |

[17] | Purchase, C. F.; Goddard, S. V.; Brown, J. A., Production of anti freeze glycoproteins in cultured and wild juvenile Atlantic cod (Gadus morhua L.) in a common laboratory environment, Canadian Journal of Zoology, 79, 610-615 (2001) |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.