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Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment

Authors: Jitsuro Sugie, Yasuhisa Saito and Meng Fan
Journal: Proc. Amer. Math. Soc. 139 (2011), 3475-3483
MSC (2010): Primary 34D23, 92D25; Secondary 34D05, 37B25
Published electronically: June 6, 2011
MathSciNet review: 2813379
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Abstract: A predator-prey model with prey receiving time-variation of the environment is considered. Such a system is shown to have a unique interior equilibrium that is globally asymptotically stable if the time-variation is bounded and weakly integrally positive. In particular, the result tells us that the equilibrium point can be stabilized even by nonnegative functions that make the limiting system structurally unstable. The method that is used to obtain the result is an analysis of asymptotic behavior of the solutions of an equivalent system to the predator-prey model.

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  • 1. A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 2005. MR 2146587 (2005m:93001)
  • 2. R. J. Ballieu and K. Peiffer, Attractivity of the origin for the equation $ \ddot{x} + f(t,x,\dot{x})\vert\dot{x}\vert^{\alpha}\dot{x} + g(x) = 0$, J. Math. Anal. Appl., 65 (1978), 321-332. MR 506309 (80a:34057)
  • 3. R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York-Toronto-London, 1953; (revised) Dover, Mineola-New York, 2008. MR 0061235 (15:794b)
  • 4. F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations, W. A. Benjamin, New York and Amsterdam, 1969; (revised) Dover, New York, 1989.
  • 5. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. MR 0190463 (32:7875)
  • 6. M. J. Crawley (ed.), Natural Enemies: The Population Biology of Predators, Parasites and Diseases, Blackwell Scientific, London-Edinburgh-Boston, 1992.
  • 7. J. Cronin, Differential Equations: Introduction and Qualitative Theory, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, 180, Marcel Dekker, New York-Basel-Hong Kong, 1994. MR 1275827 (95b:34001)
  • 8. G. F. Gause, The Struggle for Existence, Williams & Wilkins Co., Baltimore, 1934; (revised) Dover, New York, 2003.
  • 9. N. J. Gotelli, A Primer of Ecology, 4th ed., Sinauer, Sunderland Associates, 2008.
  • 10. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York-London, 1966. MR 0216103 (35:6938)
  • 11. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York-London-Sydney, 1969; (revised) Krieger, Malabar, 1980. MR 0419901 (54:7918)
  • 12. L. Hatvani, On the asymptotic stability by nondecrescent Ljapunov function, Nonlinear Anal., 8 (1984), 67-77. MR 732416 (85k:34117)
  • 13. L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal., 25 (1995), 991-1002. MR 1350721 (96k:34105)
  • 14. M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Princeton University Press, Princeton, NJ, 1978. MR 508052 (80d:92026)
  • 15. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed., Oxford Texts in Applied and Engineering Mathematics, 2, Oxford University Press, Oxford, 1999. MR 1743361 (2000j:34001)
  • 16. J. P. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961. MR 0132876 (24:A2712)
  • 17. A. J. Lotka, Elements of Physical Biology, Williams & Wilkins Co., Baltimore, 1926.
  • 18. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973.
  • 19. D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics, 24, Springer-Verlag, New York-Berlin-Heidelberg, 1997. MR 1418401 (98f:34074)
  • 20. A. N. Michel, L. Hou and D. Liu, Stability Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Birkhäuser, Boston-Basel-Berlin, 2008. MR 2351563 (2008i:93001)
  • 21. L. D. Mueller and A. Joshi, Stability in Model Populations, Princeton University Press, Princeton, NJ, 2000.
  • 22. W. W. Murdoch and A. Oaten, Predation and population stability, Adv. Ecol. Res., 9 (1975), 1-131.
  • 23. A. J. Nicholson and V. A. Bailey, The balance of animal populations, Proc. Zool. Soc. London, 1 (1935), 551-598.
  • 24. O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Zeits., 32 (1930), 703-728. MR 1545194
  • 25. K. P. Persidski, Über die Stabilität einer Bewegung nach der ersten Näherung, Mat. Sb., 40 (1933), 284-293.
  • 26. N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Applied Mathematical Sciences, 22, Springer-Verlag, New York-Heidelberg-Berlin, 1977. MR 0450715 (56:9008)
  • 27. J. Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull. Austral. Math. Soc., 78 (2008), 445-462. MR 2472280 (2009k:34102)
  • 28. J. Sugie, Influence of anti-diagonals on the asymptotic stability for linear differential systems, Monatsh. Math., 157 (2009), 163-176. MR 2504784 (2010d:34103)
  • 29. J. Sugie and Y. Ogami, Asymptotic stability for three-dimensional linear differential systems with time-varying coefficients, Quart. Appl. Math., 67 (2009), 687-705. MR 2588230 (2010k:34141)
  • 30. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York-Berlin-Heidelberg, 1990. MR 1036522 (91b:34002)
  • 31. V. Volterra, Leçons sur la Théorie Mathématique de la Lutte pour la Vie, Gauthier-Villars, Paris, 1931. Reprinted in Les Grands Classiques Gauthier-Villars, 1990. MR 1189803 (93k:92011)
  • 32. H. K. Wilson, Ordinary Differential Equations, Introductory and Intermediate Courses Using Matrix Methods, Addison-Wesley, Massachusetts-California-London-Ontario, 1971. MR 0280764 (43:6483)
  • 33. T. Yoshizawa, Stability Theory by Liapunov's Second Method, Math. Soc. Japan, Tokyo, 1966. MR 0208086 (34:7896)
  • 34. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, 14, Springer-Verlag, New York-Heidelberg-Berlin, 1975. MR 0466797 (57:6673)

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Yasuhisa Saito
Affiliation: Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
Address at time of publication: Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea

Meng Fan
Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024 Jilin, People’s Republic of China

Keywords: Global asymptotic stability, predator-prey systems, weakly integrally positive, time-variation
Received by editor(s): June 12, 2010
Published electronically: June 6, 2011
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research No.22540190 from the Japan Society for the Promotion of Science
The third author was supported in part by the NSFC, NCET-08-0755 and FRFCU
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society

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