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On a periodic predator-prey system with time delays on time scales. (English) Zbl 1221.34179
Summary: With the help of a continuation theorem based on [{\it R. Gaines} and {\it J. L. Mawhin}, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics 568, Springer-Verlag (1977; Zbl 0339.47031)] coincidence degree theory, easily verifiable criteria are established for the global existence of positive periodic solutions of the predator-prey system with time delays on time scales, given by $$ \cases &x_1^\Delta(t)=a_1(t)-b_1(t)\exp\{x_1(t-\tau_1(t))\}- \tfrac{c(t)\exp\{x_2(t-\tau_2(t))\}}{1-m\exp\{x_1(t)\}},\\ &x_2^\Delta(t)=-a_2(t)+\tfrac{b_2(t)\exp\{x_1(t-\tau_2(t))\}} {1-m\exp\{x_1(t-\tau_2(t))\}},\endcases $$ where $a_i,b_i,c,\tau_i\in C(\Bbb T,\Bbb R^+)$, $i=1,2$, are $T$-periodic functions.

34K13Periodic solutions of functional differential equations
34N05Dynamic equations on time scales or measure chains
92D25Population dynamics (general)
Full Text: DOI
[1] He, H. Z.: Stability and delays in a predator -- prey system, J math anal appl 198, 335-370 (1996) · Zbl 0873.34062 · doi:10.1006/jmaa.1996.0087
[2] Li, Y.: Periodic solutions of a periodic delay predator -- prey system, Proc amer math soc 127, 1331-1335 (1999) · Zbl 0917.34057 · doi:10.1090/S0002-9939-99-05210-7
[3] Xu, R.; Ma, Z.: Stability and Hopf bifurcation in a ratio-dependent predator -- prey system with stage structure, Chaos, solitons and fractals 38, 669-684 (2008) · Zbl 1146.34323 · doi:10.1016/j.chaos.2007.01.019
[4] 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 · doi:10.1016/j.na.2007.02.007
[5] Zhang, W.; Zhu, D.; Bi, P.: Multiple positive periodic solutions of a delayed discrete predator -- prey system with type IV functional responses, Appl math lett 20, 1031-1038 (2007) · Zbl 1142.39015 · doi:10.1016/j.aml.2006.11.005
[6] Zhu, H.; Wang, K.; Li, X.: Existence and global stability of positive periodic solutions for predator -- prey system with infinite delay and diffusion, Nonlinear anal: real world appl 8, 872-886 (2007) · Zbl 1144.34049 · doi:10.1016/j.nonrwa.2006.03.011
[7] Guo, H.; Chen, L.: The effects of impulsive harvest on a predator -- prey system with distributed time delay, Commun nonlinear sci numer simulat 14, 2301-2309 (2009) · Zbl 1221.34218 · doi:10.1016/j.cnsns.2008.05.010
[8] Zhao, C. J.: On a periodic predator -- prey system with time delays, J math anal appl 331, 978-985 (2007) · Zbl 1140.34423 · doi:10.1016/j.jmaa.2006.09.018
[9] Bohner, M.; Peterson, A.: Dynamic equation on time scales, An introduction with applications (2001) · Zbl 0978.39001
[10] Kaufmann, E. R.; Raffoul, Y. N.: Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J math anal appl 319, 315-325 (2006) · Zbl 1096.34057 · doi:10.1016/j.jmaa.2006.01.063
[11] 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 · doi:10.1016/j.nonrwa.2005.11.002
[12] Agarwal, R.; Bohner, M.; Peterson, A.: Inequalities on time scales: a survey, Math ineq appl 4, No. 4, 535-557 (2001) · Zbl 1021.34005
[13] Gaines; Mawhin, J. L.: Coincidence degree and nonlinear differential equations, (1977) · Zbl 0339.47031
[14] Agiza, H. N.; Elabbasy, E. M.; El-Metwally, H.; Elsadany, A. A.: Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear anal.: real world appl. 10, 116-129 (2009) · Zbl 1154.37335 · doi:10.1016/j.nonrwa.2007.08.029
[15] Wang, F. H.; Yeh, C. C.; Yu, S. L.; Hong, C. H.: Youngs inequality and related results on time scales, Appl math lett 18, 983-998 (2005) · Zbl 1080.26025 · doi:10.1016/j.aml.2004.06.028
[16] Xing, Y.; Han, M.; Zheng, G.: Initial value problem for first-order integro-differential equation of Volterra type on time scale, Nonlinear anal 60, 429-442 (2005) · Zbl 1065.45005 · doi:10.1016/j.na.2004.09.020