Positive Periodic Solutions in a Discrete Competitive System with the Effect of Toxic Substances
In this paper, a delayed competitive system with the effect of toxic substances is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous non-autonomous delayed competitive system with the effect of toxic substances is proposed. By using Gaines and Mawhin,s continuation theorem of coincidence degree theory, a easily verifiable sufficient condition for the existence of positive solutions of difference equations is obtained.
[1] S. Ahmad and I. M. Stamova, Asymptotic stability of an N-dimensional
impulsive competitive system, Nonlinear Anal.: Real World Appl. 8 (2)
(2007) 654-663.
[2] Y. M. Chen and Z. Zhou, Stable periodic of a discrete periodic Lotka-
Volterra competition system. J. Math. Anal. Appl. 277 (1) (2003) 358-366.
[3] M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous
ratio-dependent predator-prey system. Math. Comput. Modelling 35 (9-
10) (2002) 951-961.
[4] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear
Differential Equations. Springer-verlag, Berlin, 1997.
[5] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations
of Population Dynamics. Kluwer Academic, Boston, 1992.
[6] Y. K. Li, Positive periodic solutions of a discrete mutualism model with
time delays. Int. J. Comput. Math. Sci. 2005 (4) (2011) 499-506.
[7] Z. J. Liu, J. Hui and J. H. Wu, Permanence and partial extinction in an
impulsive delay competitive system with the effect of toxic substances. J.
Math. Chem. 46 (4) (2009) 1213-1231.
[8] L. F. Nie, Z. D. Teng, L. Hu and J. G. Peng, Existence and stability
of periodic solution of a predatorCprey model with state-dependent
impulsive effects. Math. Comput. Simul. 79 (7) (2009) 2122-2134.
[9] X. Y. Song and L. S. Chen, Periodic solutions of a delay differential
equation of plankton allelopathy. Acta. Math. Sci. Ser. A 23 (2003) 8-
13. (In Chinese)
[10] L. L. Wang and W. T. Li, Periodic solutions and permance for a delayed
nonautonomous ratio-dependent predator-prey model with Holling type
functional response. J. Comput. Appl. Math. 162 (2) (2004) 341-357.
[11] J. Wiener, Differential equations with piecewise constant delays Trends
in theory and practice of nonlinear differential equations, In Lecture Notes
in Pure and Appl. Math. Volume 90, Ddkker, New York, 1984.
[12] K. J. Zhang and Z. H. Wen, Dynamics of a discrete three species food
chain system. Int. J. Comput. Math. Sci. 5 (1) (2011) 13-15.
[13] H. Y. Zhang and Y. H. Xia, Existence of positive solutions of a discrete
time mutualism system with delays. Ann. Diff. Eqs. 22 (2) (2006) 225-
233.
[14] W. P. Zhang, D. M. Zhu and P. Bi, Multiple positive solutions of a
delayed discrete predator-prey system with type IV functional responses.
Appl. Math. Lett. 20 (10) (2007) 1031-1038.
[1] S. Ahmad and I. M. Stamova, Asymptotic stability of an N-dimensional
impulsive competitive system, Nonlinear Anal.: Real World Appl. 8 (2)
(2007) 654-663.
[2] Y. M. Chen and Z. Zhou, Stable periodic of a discrete periodic Lotka-
Volterra competition system. J. Math. Anal. Appl. 277 (1) (2003) 358-366.
[3] M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous
ratio-dependent predator-prey system. Math. Comput. Modelling 35 (9-
10) (2002) 951-961.
[4] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear
Differential Equations. Springer-verlag, Berlin, 1997.
[5] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations
of Population Dynamics. Kluwer Academic, Boston, 1992.
[6] Y. K. Li, Positive periodic solutions of a discrete mutualism model with
time delays. Int. J. Comput. Math. Sci. 2005 (4) (2011) 499-506.
[7] Z. J. Liu, J. Hui and J. H. Wu, Permanence and partial extinction in an
impulsive delay competitive system with the effect of toxic substances. J.
Math. Chem. 46 (4) (2009) 1213-1231.
[8] L. F. Nie, Z. D. Teng, L. Hu and J. G. Peng, Existence and stability
of periodic solution of a predatorCprey model with state-dependent
impulsive effects. Math. Comput. Simul. 79 (7) (2009) 2122-2134.
[9] X. Y. Song and L. S. Chen, Periodic solutions of a delay differential
equation of plankton allelopathy. Acta. Math. Sci. Ser. A 23 (2003) 8-
13. (In Chinese)
[10] L. L. Wang and W. T. Li, Periodic solutions and permance for a delayed
nonautonomous ratio-dependent predator-prey model with Holling type
functional response. J. Comput. Appl. Math. 162 (2) (2004) 341-357.
[11] J. Wiener, Differential equations with piecewise constant delays Trends
in theory and practice of nonlinear differential equations, In Lecture Notes
in Pure and Appl. Math. Volume 90, Ddkker, New York, 1984.
[12] K. J. Zhang and Z. H. Wen, Dynamics of a discrete three species food
chain system. Int. J. Comput. Math. Sci. 5 (1) (2011) 13-15.
[13] H. Y. Zhang and Y. H. Xia, Existence of positive solutions of a discrete
time mutualism system with delays. Ann. Diff. Eqs. 22 (2) (2006) 225-
233.
[14] W. P. Zhang, D. M. Zhu and P. Bi, Multiple positive solutions of a
delayed discrete predator-prey system with type IV functional responses.
Appl. Math. Lett. 20 (10) (2007) 1031-1038.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64608", author = "Changjin Xu and Qianhong Zhang", title = "Positive Periodic Solutions in a Discrete Competitive System with the Effect of Toxic Substances", abstract = "In this paper, a delayed competitive system with the effect of toxic substances is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous non-autonomous delayed competitive system with the effect of toxic substances is proposed. By using Gaines and Mawhin,s continuation theorem of coincidence degree theory, a easily verifiable sufficient condition for the existence of positive solutions of difference equations is obtained.
", keywords = "Competitive system, periodic solution, discrete time delay, topological degree.", volume = "5", number = "4", pages = "702-5", }
