## Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems.(English)Zbl 0883.34074

The authors consider the system $A\dot x_i(t)=x_i(t)F_i(t,x_1(t),\dots ,x_n(t),x_1(t-\tau (t)), \dots ,x_n(t-\tau (t)), \tag{1}$ where $$F_i$$ are periodic with respect to $$t$$ for $$n=1,\dots ,n$$. System (1) is a generalization of the nonautonomous Lotka-Volterra system, which plays a very important role in mathematical population biology. Existence and uniqueness theorems for periodic solutions of (1) are established by combining the theory of monotone flow generated by FDEs. Application to a delay nonautonomous predator-prey system is presented.

### MSC:

 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 34K13 Periodic solutions to functional-differential equations
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 [1] D. A. ARINO, T. A. BURTON AND J. R. HADDOCK, Periodic solutions to functional differential equations, Proc. Royal Soc. Edinburgh, 101A (1985), 253-271. · Zbl 0582.34077 [2] K. J. BEUTLER, C. WISSEL AND U. HALBACH, Correlation and spectral analyses of the dynamics of controlled rotifer population, Quart. Pop. Dynamics 13 (1981), 61-82. [3] G. BUTLER, H. I. FREEDMAN AND P. WALTMAN, Uniformly persistent systems, Proc. Amer. Math. Soc 96 (1986), 425^30. JSTOR: · Zbl 0603.34043 [4] G. BUTLER AND P. WALTMAN, Persistence in dynamical systems, J. Differential Equations 63 (1986), 255-263. · Zbl 0603.58033 [5] H. I. FREEDMAN AND J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992), 689-701. · Zbl 0764.92016 [6] K. GOPALSAMY, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publisher, Dordrecht, Netherlands, 1992. · Zbl 0752.34039 [7] J. R. HADDOCK AND Y. KUANG, Asymptotic theory for a class of nonautonomous delay differentia equations, J. Math. Anal. Appl. 168 (1992), 147-162. · Zbl 0764.45005 [8] U. HALBACH, Einfluss der Temperatur auf die Populationsdynamik des Planktischen Rdertiere Brachionus Calyciflorus Pallas, Decologia (Berl.) 4 (1970), 176-207. [9] U. HALBACH, Life table data and population dynamics of the rotifer Brachionus Calyciflorus Palla as influenced by periodically oscillating temperature, in ”Effects of Temperature on Ectothermic Organisms” (W. Wieser, ed.), Springer-Verlag, Heidelberg, 1973, 217-228. [10] U. HALBACH, Introductory remarks: strategies in population research exemplified by rotifer populatio dynamics, Fortschr. Zool. 25 (1979), 1-27. [11] U. HALBACH, M. SIEBERT, C. WISSEL, M. KLAUS, K. BEUTLER AND M. DELION, Population dynamic of rotifers as bioassay tool for toxic effects of organic pollutants, Verh. Internat. Verein Limnob. 21 (1981), 1147-1152. [12] J. K. HALE, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., Providence, Rhode Island, 1988. · Zbl 0642.58013 [13] J. K. HALE AND P. WALTMAN, Persistence in infinite dimensional systems, SIAM J. Math. Anal. 2 (1989), 388-395. · Zbl 0692.34053 [14] M. HE, Q. HUANG AND K. WANG, Phase spaces Cg-Ch, and 0-uniform boundedness of FDE(ID), J. Math. Anal. Appl. 138 (1984), 473-490. · Zbl 0668.34076 [15] J. HOFBAUER AND K. SiGMUND, The Theory of Evolution and Dynamical Systems, London Math. Soc Student Texts 7, Cambridge, 1988. · Zbl 0678.92010 [16] W. A. HORN, Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer Math. Soc.149 (1970), 391^04. JSTOR: · Zbl 0201.46203 [17] Y. KUANG, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. · Zbl 0777.34002 [18] Y. KUANG AND H. L. SMITH, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations 103 (1993), 221-246. · Zbl 0786.34077 [19] R. H. MARTIN AND H. L. SMITH, Convergence in Lotka-Volterra systems with diffusion and delay, Proc. Workshop on Differential Equations and Applications, Retzhof, Austria, 1989, Marcel Dekker. · Zbl 0764.35115 [20] A. SEITZ AND V. HALBACH, How is the population density regulated?, Die Naturwissenschaften 6 (1973), 1-2. [21] H. L. SMITH, Monotone semiflows generated by functional differential equations, J. Differentia Equations 66 (1987), 420-442. · Zbl 0612.34067 [22] H. L. SMITH AND H. R. THIEME, Strongly order preserving semiflowsgenerated by functional differentia equations, J. Differential Equations 93 (1991), 332-363. · Zbl 0735.34065 [23] A. TINEO, On the asymptotic behavior of some population models, J. Math. Anal. Appl. 167(1992), 516-529. · Zbl 0778.92018 [24] P. WALTMAN, A brief survey of persistence in dynamical systems, in Delay Differential Equations an Dynamical Systems, (S. Busenberg and M. Martelli, eds.), Springer-Verlag, New York, 1992, 31-40. · Zbl 0756.34054 [25] C. WISSEL, K. BEUTLER AND U. HALBACH, Correlation functions for the evaluation of repeated tim series with fluctuations, ISEM J. 3 (1981), 11-29.
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