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ON THE PERIODICITY AND GLOBAL STABILITY FOR A DISCRETE DELAYED PREDATOR–PREY MODEL

Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, P. R. China

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PEILUAN LI

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, P. R. China

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https://doi.org/10.1142/S0129167X13500869Cited by:3 (Source: Crossref)

Abstract

In this paper, a discrete version of continuous non-autonomous predator–prey model with delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and globally asymptotically stability of positive periodic solution of difference equations in consideration are established. Finally, some numerical examples are given to verify the theoretical analysis.

Keywords:

AMSC: 34K20, 34C25

References

J. M. Cushing , Math. Biosci. 31 , 143 ( 1976 ) . Crossref, Web of Science, Google Scholar
B. X. Dai and J. Z. Zou , Adv. Differential Equations 2005 , 263 ( 2005 ) . Google Scholar
M. Fan and K. Wang , Math. Comput. Modelling 35 , 951 ( 2002 ) . Crossref, Web of Science, Google Scholar
M. Fazly and M. Hesaaraki , C. R. Math. Acad. Sci. Ser. I 345 , 199 ( 2007 ) . Crossref, Web of Science, Google Scholar
H. I. Freedman , Deterministic Mathematical Models in Population Ecology , Monographs and Textbooks in Pure and Applied Mathematics ( Marcel Dekker , New York , 1980 ) . Google Scholar
R. E. Gaines and J. L. Mawhin , Coincidence Degree and Nonlinear Differential Equations ( Springer-Verlag , Berlin , 1977 ) . Crossref, Google Scholar
M. Gyllenberg , P. Yan and Y. Wang , Physica D 221 , 135 ( 2006 ) . Crossref, Web of Science, Google Scholar
J. K. Hale , Functional Differential Equations ( Springer , New York , 1971 ) . Crossref, Google Scholar
A. Leung , SIAM J. Appl. Math. 46 , 281 ( 1979 ) . Crossref, Web of Science, Google Scholar
J. J. Levin and J. A. Nohel , J. Math. Anal. Appl. 8 , 31 ( 1964 ) . Crossref, Google Scholar
Y. K. Li , Int. J. Math. Math. Sci. 2005 , 499 ( 2005 ) . Crossref, Google Scholar
L. L. Liu and Z. J. Liu , Discrete Dyn. Nat. Soc. 2011 , 15 ( 2011 ) . Google Scholar
Y. G. Sun and S. H. Saker , Appl. Math. Comput. 180 , 353 ( 2006 ) . Crossref, Web of Science, Google Scholar
G. Q. Wang and S. S. Cheng , Adv. Differential Equations 2004 , 311 ( 2004 ) . Google Scholar
L. L. Wang , W. T. Li and P. H. Zhao , Adv. Differential Equations 2004 , 321 ( 2004 ) . Google Scholar
J. Wiener , Trends in Theory and Practice of Nonlinear Differential Equations , Lecture Notes in Pure and Applied Mathematics 90 ( Dekker , New York , 1984 ) . Google Scholar
X. S. Xiong and Z. Q. Zhang , Math. Comput. Modelling 48 , 333 ( 2008 ) . Crossref, Web of Science, Google Scholar
R. Xu , L. S. Chen and F. L. Hao , Appl. Math. Comput. 171 , 91 ( 2005 ) . Crossref, Web of Science, Google Scholar
J. B. Zhang and H. Fang , Adv. Differential Equations 2006 , 1 ( 2006 ) . Crossref, Web of Science, Google Scholar
R. Y. Zhang et al. , Comput. Math. Appl. 39 , 77 ( 2000 ) . Crossref, Web of Science, Google Scholar
W. P. Zhang , D. M. Zhu and P. Bi , Appl. Math. Lett. 20 , 1031 ( 2007 ) . Crossref, Web of Science, Google Scholar