We prove the existence of positive ω-periodic solutions for some “predator–prey” systems with continuous delay of the argument for the case where the parameters of these systems are specified by ω-periodic continuous positive functions.
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J. M. Cushing, “Periodic solutions of Volterra’s population equation with hereditary effects,” SIAM J. Appl. Math., 31, No. 2, 251–261 (1976).
J. M. Cushing, “Predator–prey interactions with time delays,” J. Math. Biol., 3, No. 3-4, 369–380 (1976).
J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology [Russian translation], Mir, Moscow (1983).
V. I. Borzdyko, “Existence of a positive periodic solution for the Wangerski–Cunningham ‘predator–prey’ system with regard for periodic variations of the environment,” Differents. Uravn., 38, No. 3, 291–297 (2002).
V. I. Borzdyko, “On one topological method for proving the existence of positive periodic solutions of functional-differential equations,” Differents. Uravn., 26, No. 10, 1671–1678 (1990).
M. A. Krasnosel’skii, “Alternative principle for the existence of periodic solutions of differential equations with delayed argument,” Dokl. Akad. Nauk SSSR, 152, No. 4, 801–804 (1963).
V. I. Borzdyko, “Application of topological methods in the theory of positive periodic solutions of functional-differential equations,” Izv. Akad. Nauk Tadzh. SSR. Otd. Fiz.-Mat. Geol.-Khim. Nauk, No. 2(72), 22–30 (1979).
M. Kwapisz, “On the existence and uniqueness of solutions of differential equations with delayed argument in Banach spaces,” in: Proc. of the Sem. on the Theory of Equations with Deviating Argument [in Russian], Moscow (1967), pp. 96–110.
M. A. Krasnosel’skii, Positive Solutions of Operator Equations [in Russian], Fizmatgiz, Moscow (1962).
M. A. Krasnosel’skii, Operator of Shift Along the Trajectories of Differential Equations [in Russian], Nauka, Moscow (1966).
A. D. Myshkis, Linear Differential Equations with Delayed Argument [in Russian], Nauka, Moscow (1972).
I. P. Natanson, Theory of Functions of Real Variable [in Russian], Gostekhteorizdat, Moscow (1957).
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, Princeton (1974).
Yongkun Li, “Periodic solutions of a periodic delay predator–prey system,” Proc. Amer. Math. Soc., 127, No. 5, 1331–1335 (1999).
P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 1, pp. 15–28, January, 2010.
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Borzdyko, V.I. Periodic solutions of “predator–prey” systems with continuous delay and periodic coefficients. Ukr Math J 62, 15–30 (2010). https://doi.org/10.1007/s11253-010-0330-x
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DOI: https://doi.org/10.1007/s11253-010-0330-x