Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30172
Periodic Solutions in a Delayed Competitive System with the Effect of Toxic Substances on Time Scales

Authors: Changjin Xu, Qianhong Zhang

Abstract:

In this paper, the existence of periodic solutions of a delayed competitive system with the effect of toxic substances is investigated by using the Gaines and Mawhin,s continuation theorem of coincidence degree theory on time scales. New sufficient conditions are obtained for the existence of periodic solutions. The approach is unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations. Moreover, The approach has been widely applied to study existence of periodic solutions in differential equations and difference equations.

Keywords: Time scales, competitive system, periodic solution, coincidence degree, topological degree.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1084760

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1051

References:


[1] V. Voltera, Opere matematiche: mmemorie e note, vol. V, Acc. Naz. dei Lincei, Roma, Cremon, 1962.
[2] A.J. Lotka, Elements of Mathematical Biology, Dover, New York, 1962.
[3] W. Ding, M. Han, Dynamic of a non-autonomous predator-prey system with infinite delay and diffusion, Comput. Math. Appl., 56 (2008) 1335- 1350.
[4] Z.J. Liu, R.H. Tan, Y.P. Chen, L.S. Chen, On the stable periodic solutions of a delayed two-species model of facultative mutualism, Appl. Math. Comput., 196 (2008) 105-117.
[5] D. Summers, C. Justian, H. Brian, Chaos in periodically forced discretetime ecosystem models, Chaos, Solitons & Frctals, 11 (2000) 2331-2342.
[6] M. Fan, K. Wang, Periodic solutions of a discrete time non-autonomous ratio-dependent predator-prey system, Math. Comput. Model. 35 (2002) 951-961.
[7] W. P. Zhang, D. M. Zhu, P. Bi, Multiple positive solutions of a delayed discrete predator-prey system with type IV functional responses, Appl. Math. Lett. 20 (2007) 1031-1038.
[8] X.Y. Song, L.S. Chen, Periodic solutions of a delay differential equation of plankton allelopathy. Acta. Math. Sci. Ser. A , 23 (2003) 8-13. (in Chinese)
[9] M. Bohner, A. Peterson, Dynamic Equations on Times Scales: An Introduction with Applications, Birkh¨auser, Boston, 2001.
[10] V.Lakshmikantham, S.Sivasundaram, B.Kaymarkcalan, Dyanmic System on Measure Chains, Kluwer Academic Publishers, Boston, 1996.
[11] B.Aulbach, S.Hilger, Linear Dynamical Processes with Inhomogeneous Time Scales, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlage, Berlin, 1990.
[12] S.Hilger, Analysis on measure chains-a unfified approach to continuous and discrete calculus, Results Math. 18 (1990) 18-56.
[13] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
[14] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Berlin, Springer-verlag, 1997.