Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T07:18:39.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Global attractivity in time-delayed predator-prey systems

Published online by Cambridge University Press:  17 February 2009

Yulin Cao  and
H. I. Freedman
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a predator-prey model with time-delay due to gestation, criteria are obtained for persistence and global attractivity. The global attractivity criteria apply only to models with a decreasing prey isocline.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 2 , October 1996 , pp. 149 - 162
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Butler, G. J. and Waltman, P., “Uniformly persistent systems”, Proc. Amer. Math. Soc. 96 (1986) 425–430.CrossRefGoogle Scholar
[2]Butler, G. J., Freedman, H. I. and Waltman, P., “Persistence in dynamical systems”, J. Differential Equations 63 (1986) 255–263.CrossRefGoogle Scholar
[3]Cao, Y., Fan, J.-P and Gard, T. C., “Uniform persistence for population interaction models with time delay”, Appl. Anal. 51 (1993) 197–210.CrossRefGoogle Scholar
[4]Cao, Y. and Gard, T. C., “Extinction in predator-prey models with time delay”, Math. Biosci 118 (1993) 197–210.CrossRefGoogle ScholarPubMed
[5]Cao, Y. and Gard, T. C., “Uniform persistence for population models with time delay using multiple Lyapunov functions”, J. Diff. Integr. Equations 6 (1993) 883–898.Google Scholar
[6]Cao, Y. and Gard, T. C., “Ultimate bounds and global asymptotic stability for differential delay equations”, Rocky Mountain J. Math 25 (1995) 119–131.CrossRefGoogle Scholar
[7]Cheng, K.-S., Hsu, S.-B and Lin, S.-S., “Some results on global stability of a predator-prey system”, J. Math. Biol. 12 (1981) 115–126.CrossRefGoogle Scholar
[8]Freedman, H. I., Deterministic Mathematical Models in Population Ecology (Marcel Dekker, New York, 1980).Google Scholar
[9]Freedman, H. I. and Gopalsamy, K., “Global stability in time-delayed single species dynamics”, Bull. Math. Biol. 48 (1986) 485–192.CrossRefGoogle ScholarPubMed
[10]Freedman, H. I. and Moson, P., “Persistence definitions and their connections”, Proc. Amer. Math. Soc. 109 (1990) 1025–1033.CrossRefGoogle Scholar
[11]Freedman, H. I. and Rao, V. S. H., “The tradeoff between mutual interference and time lags in predator-prey systems”, Bull. Math. Biol. 45 (1983) 991–1004.CrossRefGoogle Scholar
[12]Freedman, H. I. and Ruan, S., “Uniform persistence in functional differential equations”, J. Differential Equations 115 (1995) 173–192.CrossRefGoogle Scholar
[13]Freedman, H. I. and Waltman, P., “Persistence in models of three interacting predator-prey populations”, Math. Biosci. 68 (1984) 213–231.CrossRefGoogle Scholar
[14]Goopalsamy, K., “Time lags and global stability in two species competition”, Bull. Math. Biol. 42 (1980) 729–737.CrossRefGoogle Scholar
[15]Hale, J. K., Theory of Functional Differential Equations (Springer Verlag, New York, 1977).CrossRefGoogle Scholar
[16]Hale, J. K. and Waltman, P., “Persistence in infinite dimensional systems”, SIAM J. Math. Anal. 20 (1989) 388–395.CrossRefGoogle Scholar
[17]Hsu, S.-B, “On global stability of a predator-prey system”, Math. Biosci 39 (1978) 1–10.CrossRefGoogle Scholar
[18]Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993).Google Scholar
[19]Liou, L. P. and Cheng, K.-S, “Global stability of a predator-prey system”, J. Math. Biol. 26 (1988) 65–71.CrossRefGoogle Scholar
[20]Wendi, W. and Zhien, M., “Harmless delays for uniform persistence”, J. Math. Anal. Appl. 158 (1991) 256–268.CrossRefGoogle Scholar