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Periodic solutions of delayed ratio-dependent predator--prey models with monotonic or nonmonotonic functional responses. (English) Zbl 1069.34098
The authors consider the following delayed ratio-dependent predator-prey system \aligned x(t) &= x(t)\Biggl[a(t)- b(t) \int^t_{-\infty} K(t- s) x(s)\,ds\Biggr]- e(t) g\Biggl({x(t)\over y(t)}\Biggr),\\ y(t) &= y(t)\Biggl[e(t) g\Biggl({x(t)- \tau(t)\over y(t)- \tau(t)}\Biggr)- d(t)\Biggr],\endaligned\tag{*} where $x(t)$ and $y(t)$ represent the predator and prey densities, respectively, $a(t)$, $b(t)$, $c(t)$, $d(t)$, $e(t)$ and $\tau(t)$ are positive periodic continuous functions with period $\omega> 0$, $\omega$ is a positive real constant. $K(s): \bbfR^+\to \bbfR^+$ is a measurable, $\omega$-periodic, normalized function such that $\int^{+\infty}_0 K(s)\,ds= 1$. By using the continuation theorem of the coincidence degree theory [see {\it R. E. Gaines} and {\it J. L. Mawhin}, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], the authors establish two main theorems on the existence of at least one positive $\omega$-periodic solution of system $(*)$ when the functional response function $g$ is monotonic or nonmonotonic. As corollaries, some applications are listed.

##### MSC:
 34K13 Periodic solutions 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 inhibitory substrates. Biotechnol. bioeng. 10, 707-723 (1986) [2] Arditi, R.; Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544-1551 (1992) [3] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator--prey systems. Nonlinear anal. TMA 32, 381-408 (1998) · Zbl 0946.34061 [4] Berryman, A. A.: The origins and evolution of predator--prey theory. Ecology 75, 1530-1535 (1992) [5] Bush, A. W.; Cook, A. E.: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. theoret. Biol. 63, 385-395 (1976) [6] Caperson, J.: Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50, 188-192 (1969) [7] Fan, M.; Wang, K.: Periodicity in a delayed ratio-dependent predator--prey system. J. math. Anal. appl. 262, 179-190 (2001) · Zbl 0994.34058 [8] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031 [9] Gutierrez, A. P.: The physiological basis of ratio-dependent predator--prey theorya metabolic pool model of Nicholson’s blowflies as an example. Ecology 73, 1552-1563 (1992) [10] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Global analysis of the michaelis--menten type ratio-dependent predator--prey system. J. math. Biol. 42, 489-506 (2001) · Zbl 0984.92035 [11] H.F. Huo, W.T. Li, Periodic solutions of a ratio-dependent food chain model with delays, Taiwanese J. Math., in press. · Zbl 1064.34045 [12] H.F. Huo, W.T. Li, Periodic solution of a periodic two-species competition model with delays, Internat. J. Appl. Math., in press. · Zbl 1043.34074 [13] 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 [14] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator--prey system. J. math. Biol. 36, 389-406 (1998) · Zbl 0895.92032 [15] Li, Y.: Periodic solutions of a periodic delay predator--prey system. Proc. amer. Math. soc. 127, 1331-1335 (1999) · Zbl 0917.34057 [16] Li, Y.; Kuang, Y.: Periodic solutions of periodic delay Lotka--Volterra equations and systems. J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062 [17] 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 [18] Sokol, W.; Howell, J. A.: Kinetics of phenol oxidation by washed cells. Biotechnol. bioeng. 23, 2039-2049 (1980) [19] L.L. Wang, W.T. Li, Existence and global stability of positive periodic solutions of a predator--prey system with delays, Appl. Math. Comput., in press. · Zbl 1029.92025 [20] 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 [21] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator--prey systems. Nonlinear anal. TMA 28, 1373-1394 (1997) · Zbl 0872.34047