We establish sufficient conditions for the existence and asymptotic stability of positive piecewise continuous almost periodic solutions to the Lotka–Volterra systems of differential equations with diffusion and pulsed action.
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M. U. Akhmet, M. Beklioglu, T. Ergenc, and V. I. Tkachenko, “An impulsive ratio-dependent predator-prey system with diffusion,” Nonlin. Anal.: Real World Appl., 7, No. 5, 1255–1267 (2006).
A. I. Dvirnyj and V. I. Slyn’ko, “Stability in terms of two measures for a class of semilinear impulsive parabolic equations,” Sb. Math., 204, No. 4, 485–507 (2013).
C. Li, X. Guo, and D. He, “An impulsive diffusion predator-prey system in three species with Beddington–DeAngelis response,” J. Appl. Math. Comput., 43, No. 1-2, 235–248 (2013).
O. O. Struk and V. I. Tkachenko, “On impulsive Lotka–Volterra systems with diffusion,” Ukr. Mat. Zh., 54, No. 4, 514–526 (2002); English translation : Ukr. Math. J., 54, No. 4, 629–646 (2002).
X. Wang and Z. Li, “Global attractivity and oscillations in a nonlinear impulsive parabolic equation with delay,” Kyungpook Math. J., 48, No. 4, 593–611 (2008).
A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri [Russian translation], Mir, Moscow (1971).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).
M. U. Akhmetov and N. A. Perestyuk, “Periodic and almost periodic solutions of strongly nonlinear impulse systems,” J. Appl. Math. Mech., 56, No. 6, 829–837 (1992).
A.V. Dvornyk and V. I. Tkachenko, “Almost periodic solutions for systems with delay and nonfixed times of impulsive actions,” Ukr. Mat. Zh., 68, No. 11, 1450–1466 (2016); English translation : 68, No. 11, 1673–1693 (2017).
R. Hakl, M. Pinto, V. Tkachenko, and S. Trofimchuk, “Almost periodic evolution systems with impulse action at state-dependent moments,” J. Math. Anal. Appl., 446, No. 1, 1030–1045 (2017).
M. Pinto and G. Robledo, “Existence and stability of almost periodic solutions in impulsive neural network models,” Appl. Math. Comput., 217, No. 8, 4167–4177 (2010).
A. M. Samoilenko and S. I. Trofimchuk, “Almost periodic impulsive systems,” Different. Equat., 29, No. 4, 684–691 (1993).
G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations, Springer, Heidelberg (2012).
V. Tkachenko, “Almost periodic solutions of parabolic type equations with impulsive action,” Funct. Different. Equat., 21, No. 3–4, 155–169 (2014).
V. Tkachenko, “Almost periodic solutions of evolution differential equations with impulsive action,” in: Mathematical Modeling and Applications in Nonlinear Dynamics, Springer, New York (2016), pp. 161–205.
W. A. Coppel, “Almost periodic properties of ordinary differential equations,” Ann. Mat. Pura Appl., Ser. 4, 76, No. 1, 27–49 (1967).
T. Yoshizawa, “Asymptotically almost periodic solutions of an almost periodic system,” Funkc. Ekvacioj., 12, No. 1, 23–40 (1969).
Yu. M. Myslo and V. I. Tkachenko, “Global attractivity in almost periodic single-species models,” Funct. Different. Equat., 18, No. 3–4, 269–278 (2011).
A. M. Samoilenko and S. I. Trofimchuk, “Unbounded functions with almost periodic differences,” Ukr. Mat. Zh., 43, No. 10, 1409–1413 (1991); English translation : Ukr. Math. J., 43, No. 10, 1306–1309 (1991).
A. V. Dvornyk and V. I. Tkachenko, “On the stability of solutions of evolutionary equations with nonfixed times of pulse actions,” Nelin. Kolyv., 18, No. 4, 475–488 (2015); English translation : J. Math. Sci., 220, No. 4, 425–439 (2017).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
L. H. Smith, Dynamics of Competition, Springer, Berlin (1999).
N. D. Alikakos, “An application of the invariance principle to reaction-diffusion equations,” J. Different. Equat., 33, No. 2, 201–225 (1979).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 2, pp. 177–192, February, 2018.
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Dvornyk, A., Struk, O.O. & Tkachenko, V.I. Almost Periodic Solutions of Lotka–Volterra Systems with Diffusion and Pulsed Action. Ukr Math J 70, 197–216 (2018). https://doi.org/10.1007/s11253-018-1495-y
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DOI: https://doi.org/10.1007/s11253-018-1495-y