×

Periodic solutions for predator-prey systems with Beddington-DeAngelis functional response on time scales. (English) Zbl 1145.92035

Summary: This paper deals with the question of existence of periodic solutions of nonautonomous predator-prey dynamical systems with Beddington-DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented by M. Bohner, M. Fan and J. Zhang [Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear. Anal. Real World Appl. 7, No. 5, 1193–1204 (2006; Zbl 1104.92057)]; M. Fan and Y. Kuang [Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelies functional response. J. Math. Anal. Appl. 295, No. 1, 15–39 (2004; Zbl 1051.34033)] and J. Zhang and J. Wang [Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response. Appl. Math. Lett. 19, No. 12, 1361-1366 (2006; Zbl 1140.92325)].

MSC:

92D40 Ecology
39A11 Stability of difference equations (MSC2000)
37N25 Dynamical systems in biology
39A12 Discrete version of topics in analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.; Bohner, M.; Peterson, A., Inequalities on time scales: a survey, Math. ineq. appl., 4, 4, 535-557, (2001) · Zbl 1021.34005
[2] Aulbach, B.; Hilger, S., Linear dynamical processes with inhomogeneous time scale, nonlinear dynamics and quantum dynamical systems, (1990), Akademie Verlage Berlin · Zbl 0723.34030
[3] Beddington, J.R., Mutual interference between parasites or predators and its effect on searching efficiency, J. animal ecol., 44, 331-340, (1975)
[4] Bohner, M.; Fan, M.; Zhang, J., Existence of periodic solutions in predator – prey and competition dynamic systems, Nonlinear anal.: real world appl., 7, 1193-1204, (2006) · Zbl 1104.92057
[5] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser New York · Zbl 0978.39001
[6] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston · Zbl 1025.34001
[7] DeAngelis, D.L.; Goldstein, R.A.; O’Neill, R.V., A model for trophic interaction, Ecology, 56, 881-892, (1975)
[8] Fan, M.; Kuang, Y., Dynamics of a nonautonomous predator – prey system with the beddington – deangelies functional response, J. math. anal. appl., 295, 15-39, (2004) · Zbl 1051.34033
[9] Gaines, R.E.; Mawhin, J.L., Coincidence degree and non-linear differential equations, (1977), Springer Berlin, MR 58:30551
[10] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[11] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Dordrecht · Zbl 0869.34039
[12] Skalski, G.T.; Gilliam, J.F., Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 11, 3083-3092, (2001)
[13] Wang, F.H.; Yeh, C.C.; Yu, S.L.; Hong, C.H., Youngs inequality and related results on time scales, Appl. math. lett., 18, 983-988, (2005) · Zbl 1080.26025
[14] Zhang, J.; Wang, J., Periodic solutions for discrete predator – prey systems with the beddington – deangelis functional response, Appl. math. lett., 19, 1361-1366, (2006) · Zbl 1140.92325
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.