# zbMATH — the first resource for mathematics

Periodic solutions in a delayed predator-prey model with nonmonotonic functional response. (English) Zbl 1172.34043
Summary: By using the continuation theorem of coincidence degree theory and some functional analytic techniques, several existence criteria are established for positive periodic solutions of a delayed predator-prey model with nonmonotonic functional response of the form
\begin{aligned} x'(t)&= x(t)(a(t)-b(t)x(t))- g(x(t))y(t),\\ y'(t)&= y(t)(\mu(t)g(x(t-\tau))-d(t)), \end{aligned}
where $$a(t)$$, $$b(t)$$, $$\mu(t)$$ and $$d(t)$$ are all positive periodic continuous functions with period $$\omega>0$$, $$\tau$$ is a nonnegative constant and $$g$$ is a nonmonotonic functional response function. And an example is given to illustrate our main result.

##### MSC:
 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general) 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] Berryman, A.A., The origins and evolution of predator – prey theory, Ecology, 75, 1530-1535, (1992) [2] Fan, Y.H.; Quan, H.S., The uniqueness theorem of limit cycle for a predator – prey system and its application, J. Lanzhou univ., 36, 6-12, (2000) · Zbl 1041.34507 [3] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023 [4] Freedman, H.I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. math. anal., 23, 689-701, (1992) · Zbl 0764.92016 [5] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer Berlin · Zbl 0339.47031 [6] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator – prey system, J. math. anal. appl., 262, 179-190, (2001) · Zbl 0994.34058 [7] Huo, H.F.; Li, W.T.; Cheng, S.S., Periodic solutions of two-species diffusive models with continuous time delays, Demonstntio math., 35, 2, 433-466, (2002) [8] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press New York [9] Li, Y.K.; Kuang, Y., Periodic solutions of periodic delay lotka – volterra equations and systems, J. math. anal. appl., 255, 260-280, (2001) · Zbl 1024.34062 [10] Ruan, S.G.; Xiao, D.M., Global analysis in a predator – prey system with nonmonotonic functional response, SIAM J. appl. math., 61, 1445-1472, (2001) · Zbl 0986.34045 [11] Smith, H.L.; Kuang, Y., Periodic solutions of delay differential equations of threshold-type delay, (), 153-176 · Zbl 0762.34044 [12] Tang, B.R.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems, Tohoku math. J., 49, 217-239, (1997) · Zbl 0883.34074 [13] Wang, L.L.; Li, W.T., Existence and global stability of positive periodic solutions of a predator – prey system with delays, Appl. math. comput., 146, 167-185, (2003) · Zbl 1029.92025 [14] Wang, L.L.; Li, W.T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator – prey model with Holling type functional response, J. comput. appl. math, 162, 321-336, (2004) [15] Wang, L.L.; Li, W.T., Existence of periodic solutions of a delayed predator – prey system with functional response, International J. math. sci., 1, 55-63, (2002) · Zbl 1075.34067 [16] Xiao, D.M.; Ruan, S.G., Multiple bifurcations in a delayed predator – prey system with nonmonotonic functional response, J. differential equations, 176, 494-510, (2001) · Zbl 1003.34064 [17] Zhao, T.; Kuang, Y.; Smith, H.L., Global existence of periodic solutions in a class of delayed Gauss-type predator – prey systems, Nonlinear anal. TMA, 28, 1373-1394, (1997) · Zbl 0872.34047
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.