×
Dai, Binxiang; Li, Ying; Luo, Zhenguo

Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses. (English) Zbl 1221.92077

Appl. Math. Comput. 217, No. 18, 7478-7487 (2011).
Summary: This paper investigates the existence of multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses and time delays. Some sufficient conditions are derived by using the continuation theorem of coincidence degree theory and analysis technique. As corollaries, some applications are listed. In particular, the presented criteria improve and extend many previous results in the literature.

Cited in 6 Documents

MSC:

92D40 Ecology
45J05 Integro-ordinary differential equations
46N60 Applications of functional analysis in biology and other sciences
45M15 Periodic solutions of integral equations

Keywords:

impulsive predator-prey system; non-monotonic functional response; multiple periodic solutions; continuation theorem
PDF BibTeX XML Cite
\textit{B. Dai} et al., Appl. Math. Comput. 217, No. 18, 7478--7487 (2011; Zbl 1221.92077)
Full Text: DOI

References:

[1] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R., Variation in plankton densities among lakes: a case for ratio-dependent models, Am. Nat., 138, 1287-1296 (1991)
[2] Gutierrez, A. P., The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson’s blowflies as an example, Ecology, 73, 1552-1563 (1992)
[3] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure, J. Math. Anal. Appl., 303, 602-621 (2005) · Zbl 1073.34090
[4] Tang, Y.; Zhang, W., Heteroclinic bifurcation in a ratio-dependent predator-prey system, J. Math. Biol., 50, 699-712 (2005) · Zbl 1067.92050
[5] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[6] Tang, S.; Chen, L., Global qualitative analysis for a ratio-dependent predator-prey model with delay, J. Math. Anal. Appl., 266, 401-419 (2002) · Zbl 1069.34122
[7] Fazly, M.; Hesaaraki, H., Periodic solutions for a discrete time predator-prey system with monotone functional response, C. R. Acad. Sci. Puris, Ser. I., 345, 199-202 (2007) · Zbl 1122.39005
[8] Gaines, R. E.; Mawkin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin
[9] Fan, Y. H.; Li, W. T.; Wang, L. L., Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonoton functional response, Nonlinear Anal. RWA, 5, 247-263 (2004) · Zbl 1069.34098
[10] Dai, B. X.; Su, H.; Hu, D. W., Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse, Nonlinear Anal., 70, 126-134 (2009) · Zbl 1166.34043
[11] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey systems, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[12] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator-prey system with non-monotonic functional response, J. Differential Equations, 176, 494-510 (2001) · Zbl 1003.34064
[13] Fazly, M.; Hesaaraki, H., Periodic solutions for predator-prey system with Beddington-DeAngelis functional response on time scales, Nonlinear Anal. RWA, 9, 1224-1235 (2008) · Zbl 1145.92035
[14] Ding, X. Q.; Lu, C.; Liu, M., Periodic solutions for a semi-ratio-dependent predator-prey system with non-monotonic functional response and time delay, Nonlinear Anal. RWA, 9, 762-775 (2008) · Zbl 1152.34046
[15] Liu, Z.; Yuan, R., Bifurcations in predatorCprey systems with nonmonotonic functional response, Nonlinear Anal. RWA, 6, 187-205 (2005) · Zbl 1089.34060
[16] Chen, Y. M., Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlinear Anal. RWA, 5, 45-53 (2004) · Zbl 1066.92050
[17] Ding, X. Q.; F Jiang, J., Multiple periodic solutions in delayed Gause-type predator-prey systems with non-monotonic numerical responses, Math. Comput. Modelling, 47, 1323-1331 (2008) · Zbl 1145.34332
[18] Smith, F. E., Population dynamics in daphnia magna and a new model for population growth, Ecology, 44, 651-663 (1963)
[19] Ballinger, G.; Liu, X., Existence, uniquence and boundness results for impulsive delay differential equations, Appl. Anal., 74, 71-93 (2000)
[20] Liu, B.; Teng, Z.; Chen, L., Analysis of a predatorCprey model with Holling II functional response concerning impulsive control strategy, J. Comput. Appl. Math., 193, 347-362 (2006) · Zbl 1089.92060
[21] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific Publishers: World Scientific Publishers Singapore · Zbl 0719.34002
[22] Bainov, D.; Simeonov, P., Impulsive Differential Equations: Periodic solutions and Applications (1993), Longman: Longman England · Zbl 0815.34001
[23] Luo, Z.; Shen, J., Impulsive stabilization of functional differential equations with infinite delays, Appl. Math. Lett., 16, 695-701 (2003) · Zbl 1068.93054
[24] Fu, X.; Yan, B.; Liu, Y., Introduction of Impulsive Differential Systems (2005), Science Press: Science Press Beijing
[25] Wang, Q.; Dai, B. X.; Chen, Y. M., Multiple periodic solutions of an impulsive predator-prey model with Holling-type IV functional response, Math. Comput. Modelling, 49, 1829-1836 (2009) · Zbl 1171.34341
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.