×

Periodic solutions for a neutral delay Hassell-Varley type predator-prey system. (English) Zbl 1410.34204

Summary: The main aim of this paper is to discuss the neutral predator-prey model with Hassell-Varley type functional response and two time-varying delays. Some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem. Finally, numerical simulations are then carried out as supporting evidences of our analytical results.

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI

References:

[1] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406, (1998) · Zbl 0895.92032
[2] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32, (1999) · Zbl 1323.92173
[3] Hsu, S.-B.; Hwang, T.-W.; Kuang, Y., Global dynamics of a predator-prey model with hassell-varley type functional response, J. Math. Biol., 10, 1-15, (2008)
[4] Hanski, I., The functional response of predator: worries bout scale, TREE, 6, 141-142, (1991)
[5] Arditi, R.; Perrin, N., H. saiah, functional response and heterogeneities: an experiment test with cladocerans, OIKOS, 60, 69-75, (1991)
[6] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[7] Frank, T. D.; Patanarapeelert, K.; Tang, I. M., Delay- and noise-induced transitions: a case study for a hongler model with time delay, Phys. Lett. A, 339, 246-251, (2005) · Zbl 1145.82319
[8] Liu, B. W., Global exponential stability for bam neural networks with time-varying delays in the leakage terms, Nonlinear Anal. RWA, 14, 559-566, (2013) · Zbl 1260.34138
[9] Zhang, G. D.; Shen, Y., Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control, Neural Netw., 55, 1-10, (2014) · Zbl 1322.93055
[10] Zhang, G. D.; Chen, B. S.; Zhu, L. L.; Shen, Y., Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput., 218, 7717-7726, (2012) · Zbl 1238.92058
[11] Kar, T. K.; Ghorai, A., Dynamic behaviour of a delayed predator-prey model with harvesting, Appl. Math. Comput., 217, 9085-9104, (2011) · Zbl 1215.92065
[12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[13] Wang, K., Periodic solutions to a delayed predator-prey model with hassell-varley type functional response, Nonlinear Anal. RWA, 12, 137-145, (2011) · Zbl 1208.34130
[14] Hassell, M.; Varley, G., New inductive population model for insect parasites and its bearing on biological control, Nature, 223, 1133-1136, (1969)
[15] Hale, J. K.; Lunel, S. M.V., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002
[16] Liu, G. R.; Yan, J. R., Existence of positive periodic solutions for neutral delay gause-type predator-prey system, Appl. Math. Model., 35, 5741-5750, (2011) · Zbl 1256.34069
[17] Liu, G. R.; Yan, J. R., Positive periodic solutions for a neutral delay ratio-dependent predator-prey model with a Holling type ii functional response, Nonlinear Anal. RWA, 12, 3252-3260, (2011) · Zbl 1231.34120
[18] Kuang, Y., On neutral delay logistic gause-type predator-prey systems, Dyn. Stab. Syst., 6, 173-189, (1991) · Zbl 0728.92016
[19] Fazly, M.; Hesaaraki, M., Periodic solutions for predator-prey systems with beddington-deangelis functional response on time scales, Nonlinear Anal. RWA, 9, 1224-1235, (2008) · Zbl 1145.92035
[20] Freedman, H. I.; Wu, J., Periodic solutions of single species models with periodic delay, SIAM J. Math. Anal. Appl., 23, 689-701, (1992) · Zbl 0764.92016
[21] Zhang, Z.; Hou, Z.; Wang, L., Multiplicity of positive periodic solutions to a generalized delayed predator-prey system with stocking, Nonlinear Anal., 68, 2608-2622, (2008) · Zbl 1146.34050
[22] Lin, G. J.; Hong, Y. G., Periodic solutions in nonautonomous predator-prey system with delays, Nonlinear Anal. RWA, 10, 1589-1600, (2009) · Zbl 1162.34306
[23] 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
[24] Fan, Y. H.; Wang, L. L., Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type iii functional response and harvesting terms, J. Math. Anal. Appl., 365, 525-540, (2010) · Zbl 1188.34112
[25] Huo, H. F.; Li, W. T., Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response, J. Comput. Appl. Math., 166, 453-463, (2004) · Zbl 1047.34081
[26] Chen, F. D., Permanence of periodic Holling type predator-prey system with stage structure for prey, Appl. Math. Comput., 182, 1849-1860, (2006) · Zbl 1111.34039
[27] Wu, X. M.; Li, J. W.; Wang, Z. C., Existence of positive periodic solutions for a generalized prey-predator model with harvesting term, Comput. Math. Appl., 55, 1895-1905, (2008) · Zbl 1242.92061
[28] Ding, X. Q.; Jiang, J. F., Positive periodic solutions in delayed gause-type predator-prey systems, J. Math. Anal. Appl., 339, 1220-1230, (2008) · Zbl 1137.34033
[29] Gaines, R.; Mawhin, J., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[30] Lu, S., On the existence of positive periodic solutions to a Lotka-Volterra cooperative population model with multiple delays, Nonlinear Anal., 68, 1746-1753, (2008) · Zbl 1139.34317
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.