×

Feedback control has no influence on the persistent property of a single species discrete model with time delays. (English) Zbl 1235.93113

Summary: By developing some new analysis techniques, we show that the following discrete single species model with feedback control and time delays is permanent. \[ \begin{cases} N(n+1)=N(n)\exp\{r(n)[1-\frac{N^2(n-m)}{k^2(n)}-c(n)u(n-\eta)]\}\\\Delta(u)=-a(n)u(n)+b(n)N(n-\sigma), \end{cases}, \] where \(N(n)\) is the density of species, \(u(n)\) is the control variable, \(\Delta \) is the first-order forward difference operator \(\Delta u(n)=u(n+1)-u(n)\). \(m,\sigma ,\eta \) are all nonnegative integers and \(r(n), a(n), k(n), c(n), b(n), d(n)\) are bounded nonnegative sequences. Our result shows that feedback control has no influence on the persistent property of the system.

MSC:

93B52 Feedback control
92D25 Population dynamics (general)
39A22 Growth, boundedness, comparison of solutions to difference equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0752.34039
[2] Fan, G.; Li, Y.; Qin, M., The existence of positive periodic solutions for periodic feedback control systems with delays, ZAMM. zeitschrift für angewandte Mathematik und mechanik, 84, 6, 425-430, (2004) · Zbl 1118.34328
[3] Chen, F.; Yang, J.; Chen, L., Note on the persistent property of a feedback control system with delays, Nonlinear analysis: real world applications, 11, 2, 1061-1066, (2010) · Zbl 1187.34106
[4] Agarwal, R.P.; Wong, P.J.Y., Advance topics in difference equations, (1997), Kluwer Publisher Dordrecht · Zbl 0914.39005
[5] Freedman, H.I., Deterministic mathematics models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[6] Murry, J.D., Mathematical biology, (1989), Springer-Verlag New York
[7] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Boston · Zbl 0752.34039
[8] Yang, X.T., Uniform persistence and periodic solutions for a discrete predator – prey system with delays, Journal of mathematical analysis and applications, 316, 1, 161-177, (2006) · Zbl 1107.39017
[9] Fan, Y.H.; Li, W.T., Permanence for a delayed discrete ratio-dependent predator – prey system with Holling type functional response, Journal of mathematical analysis and applications, 299, 2, 357-374, (2004) · Zbl 1063.39013
[10] Fan, M.; Wang, K., Periodic solutions of a discrete time nonautonomous ratio-dependent predator – prey system, Mathematical and computer modelling, 35, 9-10, 951-961, (2002) · Zbl 1050.39022
[11] Fan, M.; Wong, Patrica J.Y.; Agarwal, Ravi P., Periodicity and stability in periodic n-species lotka – volterra competition system with feedback controls and deviating arguments, Acta Mathematica sinica, 19, 4, 801-822, (2003) · Zbl 1047.34080
[12] Huo, H.; Li, W., Positive periodic solutions of a class of delay differential system with feedback control, Applied mathematics and computation, 148, 1, 35-46, (2004) · Zbl 1057.34093
[13] Chen, F.D.; Liao, X.Y.; Huang, Z.K., The dynamic behavior of N-species cooperation system with continuous timedelays and feedback controls, Applied mathematics and computation, 181, 2, 803-815, (2006) · Zbl 1102.93021
[14] Chen, F.D., Permanence of a single species discrete model with feedback control and delay, Applied mathematics letters, 20, 7, 729-733, (2007) · Zbl 1128.92029
[15] Y.H. Fan, L.L. Wang, Permanence for a discrete model with feedback control and delay, Discrete Dynamics in Nature and Society, vol. 2008, Article ID 945109, 8 p., doi:10.1155/2008/945109.
[16] Wang, L.; Wang, M.Q., Ordinary difference equation, (1991), Xinjiang University Press China, (in Chinese)
[17] Chen, F.D., Permanence and global attractivity of a discrete multispecies lotka – volterra competition predator – prey systems, Applied mathematics and computation, 182, 1, 3-12, (2006) · Zbl 1113.92061
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.