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Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. (English) Zbl 1065.92066
Summary: The principle aim of this paper is to explore the existence of periodic solutions with strictly positive components of generalized ecological competition systems governed by impulsive differential equations with infinite delays. Easily verifiable sufficient criteria are established. The approach is based on the coincidence degree theory and its related continuation theorem as well as some a priori estimates. Applications to some famous competition models, which have been widely studied in the literature, are presented also.

34K13Periodic solutions of functional differential equations
34A37Differential equations with impulses
34K60Qualitative investigation and simulation of models
Full Text: DOI
[1] Samoilenko, A. M.; Perestyuk, N. A.: Differential equations with impulse effect. (1987)
[2] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[3] Anokhin, A. V.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. math. Anal. appl. 193, 923-941 (1995) · Zbl 0837.34076
[4] Bainov, D. D.; Covachev, V.; Stamova, I.: Stability under persistent disturbances of impulsive differential-difference equations of neutral type. J. math. Anal. appl. 187, 790-808 (1994) · Zbl 0811.34057
[5] Shen, J. H.: On some asymptotic stability results of impulsive integro-differential equations. Chinese math. Ann. 17A, 759-765 (1996) · Zbl 0877.34051
[6] Shen, J. H.: The existence of non-oscillatory solutions of delay differential equations with impulses. Appl. math. Comput. 77, 156-165 (1996) · Zbl 0861.34044
[7] Shen, J. H.; Yan, J. R.: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear anal. TMA 33, 519-531 (1998) · Zbl 0933.34083
[8] Yu, J. S.; Zhang, B. G.: Stability theorems for delay differential equations with impulses. J. math. Anal. appl. 199, 162-175 (1996) · Zbl 0853.34068
[9] Gopalsamy, K.: Global asymptotic stability in Volterra’s population systems. J. math. Biol. 19, 157-168 (1984) · Zbl 0535.92020
[10] Gopalsamy, K.: Stability and oscillation in delay differential equations of population dynamics. Mathematics and its applications 74 (1992) · Zbl 0752.34039
[11] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[12] Kuang, Y.; Smith, H. L.: Global stability for infinite delay Lotka-Volterra type systems. J. diff. Eqs. 103, 221-246 (1993) · Zbl 0786.34077
[13] Leung, A. W.; Zhou, Z.: Global stability for a large class of Volterra-Lotka type integro-differential population delay equations. Nonlinear anal. TMA 12, 495-505 (1988) · Zbl 0651.45004
[14] Fan, M.; Wang, K.: Existence of positive periodic solution of a class of integro-differential equations. Acta Mathematica sinica 44, 437-444 (2001) · Zbl 1010.45002
[15] Jin, Z.: The study for ecological and epidemical models influenced by impulses. Doctoral thesis (2001)
[16] Gaines, R. E.; Mawhin, J. L.: Coincidence degree, and nonlinear differential equations. (1977) · Zbl 0339.47031
[17] Gilpin, M. E.; Ayala, F. J.: Global models of growth and competition. Proc. nat. Acad. sci. USA 70, 3590-3593 (1973) · Zbl 0272.92016
[18] Ayala, F. J.; Gilpin, M. E.; Eherenfeld, J. G.: Competition between species: theoretical models and experimental tests. Theoretical population biology 4, 331-356 (1973)
[19] Fan, M.; Wang, K.: Global periodic solutions of generalized n-species gilpin-ayala competition model. Computers math. Applic. 40, No. 10/11, 1141-1151 (2000) · Zbl 0954.92027
[20] Gilpin, M. E.; Ayala, F. J.: Schoener’s model and drosophila competition. Theoretical population biology 9, 12-14 (1976)
[21] Goh, B. S.; Agnew, T. T.: Stability in gilpin and ayala’s model of competition. J. math. Biol. 4, 275-279 (1977) · Zbl 0379.92017
[22] Liao, X.; Li, J.: Stability in gilpin-ayala competition models with diffusion. Nonliear anal. TMA 28, No. 10, 1751-1758 (1997) · Zbl 0872.35054
[23] Maynard-Smith, J.: Models in ecology. (1974) · Zbl 0312.92001
[24] Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. modelling 84, 287-289 (1996)
[25] Zhen, J.; Ma, Z. E.: Periodic solutions for delay differential equations model of plankton allelopathy. Computers math. Applic. 44, No. 3/4, 491-500 (2002) · Zbl 1094.34542
[26] Mukhopadhyay, A.; Chattopadhyay, J.; Tapaswi, P. K.: A delay differential equations model of plankton allelopathy. Mathematical biosciences 149, 167-189 (1998) · Zbl 0946.92031
[27] Beltrami, E.; Carroll, T. O.: Modeling the role of viral disease in recurrent phytoplankton blooms. J. math. Biol. 32, 857-863 (1994) · Zbl 0825.92122
[28] May, R. M.; Leonard, W. J.: Nonlinear aspects of competition between three species. SIAM J. Appl. math. 29, 243-253 (1975) · Zbl 0314.92008