Periodic solutions, global attractivity and oscillation of an impulsive delay host-macroparasite model. (English) Zbl 1165.34350

Saker, S. H.; Alzabut, J. O.

Math. Comput. Modelling 45, No. 5-6, 531-543 (2007).

Summary: We consider the nonlinear impulsive delay host-macroparasite model with periodic coefficients. By means of the continuation theorem of coincidence degree, we establish a sufficient condition for the existence of a positive periodic solution \(\bar{M}(t)\) with strictly positive components. Moreover, we establish a sufficient condition for the global attractivity of \(\bar{M}(t)\) and some sufficient conditions for oscillation of all positive solutions about the positive periodic solution \(\bar{M}(t)\).

MSC:

34C25	Periodic solutions to ordinary differential equations
34A37	Ordinary differential equations with impulses
34D45	Attractors of solutions to ordinary differential equations
92D25	Population dynamics (general)

Keywords:

impulse; delay; existence; global attractivity; oscillation; macroparasite model

References:

[1]	Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2]	Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific · Zbl 0837.34003
[3]	Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[4]	Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048
[5]	May, R.; Anderson, R. M., Infectious Diseases of Humans: Dynamics and Control (1995), Oxford Science Publications
[6]	Macdonald, N., Biological Delay Systems: Linear Stability Theory, Cambridge Studies in Mathematical Biology (1989), Cambridge Univ. Press · Zbl 0669.92001
[8]	Hairston, N. G., On the mathematical analysis of schistosome population, Bull. World Health Organ., 33, 45-62 (1965)
[9]	Macdonald, G., The dynamics of Hilmenth infectious with special references to schistosomes, Trans. R. Soc. Trop. Med. Hyg., 59, 489-506 (1965)
[10]	Tallis, G. M.; Leyton, M. K., A schistosome approach to the study of parasite populations, J. Theor. Biol., 13, 160-172 (1966)
[11]	Leyton, M. K., Stochastic models in population of Hilmenthic parasites in the definitive host. II Sexual mating functions, Math. Biosci., 39, 413-419 (1968) · Zbl 0181.47906
[12]	Chao, J., On the oscillation of linear differential equations with deviating arguments, Math. Practice Theory, 1, 32-40 (1991) · Zbl 1493.34178
[13]	Elabbasy, E. M.; Saif, K.; Saker, S. H., Oscillation in host macroparasite model with delay time, Far East J. Appl. Math., 4, 119-142 (2000) · Zbl 0992.34047
[14]	Erbe, L. H.; Zhang, B. G., Oscillation for first order linear differential equations with deviating arguments, Differ. Integral Equ., 1, 305-314 (1988) · Zbl 0723.34055
[15]	Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0326.34021
[16]	Ladde, G. S.; Lakshmikantham, V.; Zhang, B. Z., Oscillation Theory of Differential Equations with Deviating Arguments (1987), Marcel Dekker: Marcel Dekker New York · Zbl 0832.34071
[17]	Li, B., Oscillation of delay differential equations with variable coefficients, J. Math. Anal. Appl., 192, 217-234 (1995)
[18]	Luo, J., Oscillation of nonlinear delay differential equations with impulses, J. Math. Anal. Appl., 250, 290-298 (2000) · Zbl 0971.34055
[19]	Nicholson, A. J., The balance of animal population, J. Animal Ecology, 2, 132-178 (1993)
[20]	Shen, J.; Tang, X., New oscillation criteria for linear delay differential equations, Comput. Math. Appl., 36, 53-61 (1998) · Zbl 0961.34056
[21]	Yu, J. S.; Wang, Z. C., Some further results on oscillation of neutral differential equations, Bull. Austral. Math. Soc., 46, 149-157 (1992)
[22]	Yu, J. S.; Wang, Z. C.; Zhang, B. G.; Qian, X. Z., Oscillations of differential equations with deviating arguments, Panamer. Math. J., 2, 59-72 (1992) · Zbl 0845.34082