Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-06T21:01:53.081Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays

Published online by Cambridge University Press:  14 November 2011

Xue-Zhong He
Xue-Zhong He
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Get access Rights & Permissions [Opens in a new window]

Extract

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delays

is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 755 - 771
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Beretta, E. and Kuang, Y.. Convergence results in a well-known delayed predator–prey system. J. Math. Analyt. Applic. 204 (1996), 840–853.CrossRefGoogle Scholar
2Cooke, K. and Grossman, Z.. Discrete delay, distributed delay and stability switches. J. Math. Analyt. Applic. 86 (1982), 592–627.CrossRefGoogle Scholar
3Cooke, K. and Driessche, P. van den. On zeros of some transcendental equations. Funkcialaj Ekvacioj 29 (1986), 77–90.Google Scholar
4Cushing, J. M.. Integrodifferential iequations and delay models in population dynamics. Lecture Notes in Biomathematics, vol. 20 (New York: Springer, 1977).CrossRefGoogle Scholar
5Cushing, J. M.. On the oscillatory nature of solutions of general predator–prey models with time delays. Nonlinear Analysis TMA 1 (1977), 583–592.CrossRefGoogle Scholar
6Cushing, J. M.Stability and instability in predator–prey models with growth rate response delays. Rock. Mount. J. Math. 9 (1979), 43–50.Google Scholar
7El-Owaidy, H. and Ammar, A. A.. Stable oscillations in a predator–prey model with time lag. J. Math. Analyt. Applic. 130 (1988), 191–199.CrossRefGoogle Scholar
8Farkas, A., Farkas, M. and Szabo, G.. Multiparameter bifurcation diagram in predator–prey model with time lags. J. Math. Biol. 26 (1988), 93–103.CrossRefGoogle Scholar
9Farkas, M.. Stable oscillations in a predator–prey system with time delay. J. Math. Analyt. Applic. 102 (1984), 175–188.CrossRefGoogle Scholar
10Freedman, H. I. and Rao, V. S. H.. Stability criteria for a system involving two time delays. SIAM J. Appl. Math. 46 (1986), 552–560.CrossRefGoogle Scholar
11Goh, B. S.. Global stability in many species systems. Am. Nature 111 (1977), 135–143.CrossRefGoogle Scholar
12Gopalsamy, K.. Stability criteria for the linear system Ẋ(t) + A(t) X(t − т) = 0 and an application to a non-linear system. Int. J. System Sci. 21 (1990), 1841–1853.CrossRefGoogle Scholar
13Gopalsamy, K.. Stability and oscillations in delay differential equations of population dynamics (Dordrecht, The Netherlands: Kluwer, 1992).CrossRefGoogle Scholar
14Hassard, M. W., Kazarinoff, N. D. and Wan, Y. H.. Theory and applications of Hopf bifurcation. London Mathematical Society Lecture Notes Series, vol. 41 (Cambridge University Press, 1981).Google Scholar
15He, X.-Z.. Stability and delays in a predator–prey system. J. Math. Analyt. Applic. 198 (1996), 355–370.CrossRefGoogle Scholar
16He, X.-Z.. The Lyapunov functionals for delay Lotka–Volterra type models. SIAM J. Appl. Math. 58 (1998), 1222–1236.CrossRefGoogle Scholar
17He, X. and Gopalsamy, K.. Global stability in n-species competition modelled by ‘pure-delaytype’ systems. I. Autonomous case. Preprint, Department of Mathematics, The Flinders University, Australia (1995).Google Scholar
18Hofbauer, J. and Sigmund, K.. The theory of evolution and dynamical systems (Cambridge University Press, 1988).Google Scholar
19Huang, Q., Wei, J., Wu, J. and Zou, X.. Direction and stability of bifurcating periodic solutions in predator–prey systems with discrete delay. In Differential equationsand control theory (ed. Lu, G.Deng, Z., Liang, Z. and Ruan, S.), pp. 107–119. Lecture Notes in Pure and Applied Mathematics, vol. 176 (New York: Marcel Dekker, 1996).Google Scholar
20Kolmanovskii, V. B., Torelli, L. and Vermiglio, R.. Stability of some test equations withdelay. SIAM J. Math. Analysis 25 (1994), 948–961.CrossRefGoogle Scholar
21Kuang, Y.. Delay differential equations with applications in population dynamics (New York: Academic, 1993).Google Scholar
22Kuang, Y.. Global stability in delay differential systems without dominating instantaneous negative feedbacks. J. Diff Eqns 119 (1995), 503–532.CrossRefGoogle Scholar
23Kuang, Y. and Smith, H. L.. Convergence in Lotka–Volterra-type delay systems without instaneous feedbacks. Proc. R. Soc. Edinb. A 123 (1993), 45–58.CrossRefGoogle Scholar
24Kuang, Y. and Smith, H. L.. Global stability for infinite delay Lotka–Volterra type systems. J. Diff. Eqns 103 (1993), 221–246.CrossRefGoogle Scholar
25Leung, A.. Periodic solutions for a prey–predator differential delay equation. J. Diff. Eqns 26 (1977), 391–403.CrossRefGoogle Scholar
26Leung, A.. Conditions for global stability concerning a prey–predator model withdelay effects. SIAM J. Appl. Math. 36 (1979), 281—286.CrossRefGoogle Scholar
27Leung, A. and Zhou, Z.. Global stability for large systems of Volterra–Lotka typeintegrodifferential population delay equations. Nonlinear Analysis TMA 12 (1988), 495–505.CrossRefGoogle Scholar
28Shukla, V. P.. Conditions for global stability of two-species population models with discrete time delay. Bull. Math. Biol. 45 (1983), 793–805.CrossRefGoogle Scholar
29Stepan, G.. Great delay in a predator–prey model. Nonlinear Analysis TMA 10 (1986), 913–929.CrossRefGoogle Scholar
30Wörz-Busekros, A.. Global stability in ecological systems with continuous time delay. SIAM J. Appl. Math. 35 (1978), 123–134.CrossRefGoogle Scholar