Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations

Dynamic Systems and Applications

Volumn 17, Issue 1, 2008, Pages 235-253

  Shaikhet, Leonid ^a  

^a Mariupol State University   (Ukraine)

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[No Author keywords available]

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EID: 49049108886     PISSN: 10562176     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (39)

1. M. Bandyopadhyay and J. Chattopadhyay, Ratio dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18:913-936, 2005.
2. E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation (Special Issue "Delay Systems"), 45(N.3-4):269-277, 1998.
3. E. Beretta and Y. Kuang, Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32(N.4):381-408, 1998.
4. E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays: Boundedness, local and global asymptotic stability, Diff. Eqns and Dynamical Systems, 2(N.1):19-40, 1994.
5. E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays II, Diff. Eqns and Dynamical Systems, 2(N.4):263-288, 1994.
6. E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol, 33:250-260, 1995.
7. A.W. Bush and A.E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, Journal of Theoretical Biology, 63:385-395, 1976.
8. M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175:117-131, 2002.
9. F. Chen, Periodicity in a ratio-dependent predator-prey system with stage structure for predator, Journal of Applied Mathematics, 2005(N.2):153-169, 2005.
10. J.B. Colliugs, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol, 36:149-168, 1997.
11. M. Fan, Q. Wang and X. Zhou, Dynamics of a nonautonomous ratio-dependent predator-prey system, Proc. Royal Soc. Edin. A., 133:97-118, 2003.
12. Y.H. Fan, W.T. Li and L.L. Wang, Periodic solutions of delayed ratio-dependent predator-prey model with monotonic and nomonotonic functional response, Nonlinear Analysis RWA, 5:(N.2):247-263, 2004.
13. I.I. Gikhman and A.V. Skorokhod, The theory of stochastic processes, I, II, III, Springer Verlag, Berlin, 1974, 1975, 1979.
14. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992.
15. S.A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology, 49:188-200, 2004.
16. A. Hastings, Age dependent predation is not a simple process. I. Continuous time models, Theoretical Population Biology, 23:347-362, 1983.
17. A. Hastings, Delays in recruitment at different trophic levels effects on stability, Journal of Mathematical Biology, 21:35-44, 1984.
18. H.F. Huo and W.T. Li, Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response, J. Comput. Appl. Math., 166:453-463, 2004.
19. V.B. Kolmanovskii and A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Boston, USA, 1992.
20. V.B. Kolmanovskii and L.E. Shaikhet, New results in stability theory for stochastic functional-differential equations (SFDEs) and their applications, In Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc, 1:167-171, 1994.
21. V.B. Kolmanovskii and L.E. Shaikhet, A method for constructing Lyapunov functionals for stochastic differential equations of neutral type, Differentialniye uravneniya, 31(N.11):1851-1857, 1995 (in Russian).
22. Translation in Differential Equations, 31(N.11):1819-1825, 1996.
23. V.B. Kolmanovskii and L.E. Shaikhet, Some peculiarities of the general method of Lyapunov functionals construction, Applied Math. Lett., 15(N.3):355-360, 2002.
24. V.B. Kolmanovskii and L.E. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Mathematical and Computer Modelling, 36(N.6):691-716, 2002.
25. V.B. Kolmanovskii and L.E. Shaikhet, About one application of the general method of Lyapunov functionals construction, International Journal of Robust and Nonlinear Control, Special Issue on Time Delay Systems, RNC., 13(N.9):805-818, 2003.
26. Y. Kuang, Delay differential equations with application in population dynamics, Academic Press, New York, USA, 1993.
27. P.H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45:16-31, 1958.
28. Y.K. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255:260-280, 2001.
29. S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61:1445-1472, 2001.
30. L.E. Shaikhet, Modem state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of Stochastic Processes, 2(18)(N.1-2):248-259, 1996.
31. L.E. Shaikhet, Stability in probability of nonlinear stochastic systems with delay, Matematicheskiye zametki, 57(N.1):142-146, 1995 (in Russian).
32. Translation in Math. Notes, 57(N.1-2):103-106, 1995.
33. L.E. Shaikhet, Stability in probability of nonlinear stochastic hereditary systems, Dynamic Systems and Applications, 4(N.2):199-204, 1995.
34. L.E. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Applications, 1(N.1):3-13, 1998.
35. L.L. Wang and W.T. Li, Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146(N.1):167-185, 2003.
36. L.L. Wang and W.T. Li, Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response, Discrete Dynamics in Nature and Society, 2004(N.2):325-343, 2004.
37. Q. Wang, M. Fan and K. Wang, Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey system with functional responses, J. Math. Anal. Appl., 278:443-471, 2003.
38. D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations, 176:494-510, 2001.
39. X. Zhang, L. Chen and U.A. Neumann, The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences, 168:201-210, 2000.