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Periodic solutions for a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay. (English) Zbl 1152.34046
By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a semi-ratio-dependent predator-prey system with nonmonotonic functional responses and time delay is established. Further, by constructing a Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the uniqueness and global stability of positive periodic solutions to the system. Finally, some numerical simulations are carried out to illustrate the effectiveness of the new results.

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
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##### References:
 [1] Andrews, J.F., A mathematical model for the continuous culture of microorganisms utilizing inhabitory substrates, Biotechnol. bioengrg., 10, 707-723, (1986) [2] Bohner, M.; Fan, M.; Zhang, J.M., Existence of periodic solutions in predator – prey and competition dynamic systems, Nonlinear anal. real world appl., 7, 1193-1204, (2006) · Zbl 1104.92057 [3] Bush, A.W.; Cook, A.E., The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. theor. biol., 63, 385-395, (1976) [4] Chen, Y.M., Multiple periodic solutions of delayed predator – prey systems with type IV functional responses, Nonlinear anal. real world appl., 5, 45-53, (2004) · Zbl 1066.92050 [5] Chen, X.X.; Chen, F.D., Almost-periodic solutions of a delay population equation with feedback control, Nonlinear anal. real world appl., 7, 559-571, (2006) · Zbl 1128.34045 [6] Collings, J.B., The effects of the functional response on the bifurcation behavior of a mite predator – prey interaction model, J. math. biol., 36, 149-168, (1997) · Zbl 0890.92021 [7] Cushing, J.M., Periodic time-dependent predator – prey system, SIAM J. appl. math., 32, 82-95, (1977) · Zbl 0348.34031 [8] Fan, Y.H.; Li, W.T.; Wang, L.L., Periodic solutions of delayed ratio-dependent predator – prey model with monotonic and nonmonotonic functional response, Nonlinear anal. real world appl., 5, 2, 247-263, (2004) · Zbl 1069.34098 [9] Fan, M.; Wang, K., Existence and global attractivity of positive periodic solutions of periodic $$n$$-species lotka – volterra competition systems with several deviating arguments, Math. biosci., 160, 47-61, (1999) · Zbl 0964.34059 [10] Fan, M.; Wang, Q., Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator – prey system, Discrete and continuous dynam. systems B, 4, 3, 563-574, (2004) · Zbl 1100.92064 [11] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer Berlin · Zbl 0339.47031 [12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht/Norwell, MA · Zbl 0752.34039 [13] Huo, H.F., Periodic solutions for a semi-ratio-dependent predator – prey system with functional responses, Appl. math. lett., 18, 313-320, (2005) · Zbl 1079.34515 [14] Ruan, S.; Xiao, D., Global analysis in a predator – prey system with nonmonotonic functional response, SIAM J. appl. math., 61, 1445-1472, (2001) · Zbl 0986.34045 [15] Sokol, W.; Howell, J.A., Kinetics of phenol oxidation by washed cells, Biotechnol. bioeng., 23, 2039-2049, (1980) [16] Wang, Q.; Fan, M.; Wang, K., Dynamics of a class of nonautonomous semi-ratio-dependent predator – prey system with functional responses, J. math. anal. appl., 278, 443-471, (2003) · Zbl 1029.34042 [17] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator – prey system with nonmonotonic functional response, J. differential equations, 176, 494-510, (2001) · Zbl 1003.34064 [18] Xu, R.; Chaplain, M.A.J.; Davidson, F.A., Periodic solutions for a predator – prey model with Holling-type functional response and time delays, Appl. math. comput., 161, 637-654, (2005) · Zbl 1064.34053
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