The Lyapunov functionals for delay Lotka-Volterra-type models

SIAM Journal on Applied Mathematics

Volumn 58, Issue 4, 1998, Pages 1222-1236

  He, Xue Zhong ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Delays; Global attractivity; Lotka Volterra equations; Lyapunov functional

Indexed keywords

ESTIMATION; FUNCTIONS; INTEGRODIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; ORDINARY DIFFERENTIAL EQUATIONS;

LOTKA-VOLTERRA MODEL; LYAPUNOV FUNCTIONALS;

LYAPUNOV METHODS;

EID: 0032138492     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139995295116     Document Type: Article

References (35)

1. E. BERETTA AND Y. TAKEUCHI, Qualitative properties of chemostat equations with time delays II, Differential Equations Dynam. Systems, 2 (1994), pp. 263-288.
2. Y. CAO, J. FAN, AND T. G. GARD, Uniform persistence for population interaction models with delay, Appl. Anal., 51 (1993), pp. 197-210.
3. Y. CAO AND T. C. GARD, Extinction in predator-prey models with time delay, Math. Biosci., 118 (1993), pp. 197-210.
4. Y. CAO AND T. C. GARD, Uniform persistence for population models with time delay using multiple Lyapunov functions, Differential Integral Equations, 6 (1993), pp. 883-898.
5. L. CHEN, Z. LU, AND W. WANG, The effect of delays on the permanence for Lotka-Volterra systems, Appl. Math. Lett., 8 (1995), pp. 71-73.
6. J. M. GUSHING, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), pp. 82-95.
7. J. M. CUSHING, Stability and instability in predator-prey models with growth rate response delays, Rocky Mountain J. Math, 9 (1979), pp. 43-50.
8. J. M. CUSHING, Integrodifferential equations and delay models in population dynamics, Lecture Notes in Biomathematics 20, Springer-Verlag, New York, 1977.
9. H. I. FREEDMAN AND K. GOPALSAMY, Nonoccurrence of stability switching in systems with discrete delays, Canad. Math. Bull., 31 (1988), pp. 52-58.
10. H. I. FREEDMAN AND V. S. H. RAO, Stability criteria for a system involving two time delays, SIAM J. Appl. Math., 46 (1986), pp. 552-560.
11. H. I. FREEDMAN AND S. RUAN, Uniform persistence in functional differential equations, J. Differential Equations, 115 (1995), pp. 173-192.
12. H. I. FREEDMAN, J. SO, AND P. WALTMAN, Coexistence in a model of competition in the chemostat incorporating discrete delays, SIAM J. Appl. Math., 49 (1989), pp. 859-970.
13. B. S. GOH, Global stability in many species systems, Amer. Natur., 111 (1977), pp. 135-143.
14. K. GOPALSAMY, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, The Netherlands, 1992.
15. K. GOPALSAMY AND X. HE, Dynamics of an almost periodic integrodifferential equation, Methods Appl. Anal., 2 (1995), pp. 38-66.
16. K. GOPALSAMY, X. HE, AND L. WEN, Global attractivity and oscillations in a periodic logistic integrodifferential equation, Houston J. Math., 17 (1991), pp. 157-177.
17. J. HOFBAUER AND K. SIGMUND, The Theory of Evolution and Dynamical Systems, Cambridge, New York, 1988.
18. J. K. HALE, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
19. J. K. HALE AND P. WALTMAN, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), pp. 388-395.
20. P. R. HALMOS, Measure Theory, Van Nostrand, New York, 1951.
21. X. HE, Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198 (1996), pp. 355-370.
22. X. HE AND K. GOPALSAMY, Persistence, stability and level crossings in an integrodifferential system, J. Math. Biol., 32 (1994), pp. 395-426.
23. X. HE AND K. GOPALSAMY, Global stability in n-species competition modelled by "pure-delay-type" systems I: Autonomous case, preprint, School of Mathematics and Statistics, University of Sydney, Sydney, Australia, 1995.
24. V. HUSTON AND K. SCHMITT, Permanence and dynamics of biological systems, Math. Biosci., III (1992), pp. 1-71.
25. V. B. KOLMANOVSKH, L. TORELLI, AND R. VERMIGLIO, Stability of some test equations with delay, SIAM J. Math. Anal., 25 (1994), pp. 948-961.
26. Y. KUANG, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
27. Y. KUANG AND H. L. SMITH, Convergence in Lotka-Volterra-type delay systems without instantaneous feedbacks, Proc. Roy. Soc. Edinburgh, 123 Sect. A (1993), pp. 45-58.
28. A. LEUNG, Conditions for global stability concerning a prey-predator model with delay effects, SIAM J. Appl. Math., 36 (1979), pp. 281-286.
29. A. LEUNG AND Z. ZHOU, Global stability for large systems of Volterra-Lotka type integrodifferential population delay equations, Nonlinear Anal., 12 (1988), pp. 495-505.
30. Z. LU AND Y. TAKEUCHI, Permanence and global attractivity for competitive Lotka-Volterra systems with delay, Nonlinear Anal., 22 (1994), pp. 847-856.
31. V. P. SHUKLA, Conditions for global stability of two-species population models with discrete time delay, Bull. Math. Biol., 45 (1983), pp. 793-805.
32. G. STEFAN, Great delay in a predator-prey model, Nonlinear Anal., 10 (1986), pp. 913-929.
33. V. VOLTERRA, Leçons sur la théori mathématique de la Lutte pure la vie, Gauthier-Villars, Paris, 1931.
34. W. WANG AND Z. MA, Harmless delays for uniform persistence, J. Math. Anal. Appl., 158 (1991), pp. 256-268.
35. A. WÖRZ-BUSEKROS, Global stability in ecological systems with continuous time delay, SIAM J. Appl. Math., 35 (1978), pp. 123-134.