Abstract
We study a two-dimensional Lotka–Volterra system with diffusion and impulse action at fixed moments of time. We establish conditions for the permanence of the system. In the case where the coefficients of the system are periodic in t and independent of the space variable x, we obtain conditions for the existence and uniqueness of periodic solutions of the system.
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REFERENCES
T. A. Akramov and M. P. Vishnevskii, “On some qualitative properties of a reaction-diffusion system,” Sib. Mat. Zh., 36, No. 1, 3–19 (1995).
N. D. Alikakos, “An application of the invariance principle to reaction-diffusion equations,” J. Different. Equat., 33, 201–225 (1979).
C. Alvarez and A. Lazer, “Application of topological degree to the periodic competing species problem,” J. Austral. Math. Soc., Ser. B, 28, 202–219 (1986).
S. R. Dunbar, K. P. Rubakowski, and K. Schmitt, “Persistence in models of predator-prey populations with diffusion,” J. Different. Equat., 65, 117–138 (1986).
L. Dung, “Global attractors and steady state solutions for a class of reaction-diffusion systems,” J. Different. Equat., 147, 1–29 (1998).
V. Hutson and K. Schmitt, “Permanence and dynamics of biological systems,” Math. Biosci., 111, 1–71 (1992).
K. Mazuda, “On global existence and asymptotic behavior of solutions of reaction-diffusion equations,” Hokkaido Math. J., 12, 360–370 (1983).
R. Redlinger, “Compactness results for time-dependent parabolic systems,” J. Different. Equat., 64, 133–153 (1986).
L. H. Smith, “Dynamics of competition,” Lect. Notes Math., 1714, 192–240 (1999).
X. Liu, “Boundedness in terms of two measures and permanence of population growth models,” in: Nonlinear Analysis TMA: Proceedings of the Second World Congress of Nonlinear Analysis, 30, Pt. 5 (1997), pp. 2711–2723.
O. O. Struk and V. I. Tkachenko, “On two-dimensional impulsive Lotka-Volterra systems,” Nelin. Kolyvannya, 3, No. 4, 554–561 (2000).
O. O. Struk and V. I. Tkachenko, “On periodic solutions of Lotka-Volterra systems with impulses,” Nonlin. Oscillations, 4, No. 1, 129–144 (2001).
D. D. Bainov, E. Minchev, and K. Nakagawa, “Asymptotic behaviour of solutions of impulsive semilinear parabolic equations,” Nonlin. Anal. Appl., 30, 2725–2734 (1997).
C. Y. Chan, L. Ke, and A. S. Vasala, “Impulsive quenching for reaction-diffusion equations,” Nonlin. Anal. Theory Methods Appl., 22, No. 11, 1323–1328 (1994).
Yu. V. Rogovchenko and S. I. Trofimchuk, Periodic Solutions of Weakly Nonlinear Partial Differential Equations of Parabolic Type with Impulse Action [in Russian], Preprint 86.65, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986).
P. E. Sobolevskii, “On equations of parabolic type in a Banach space,” Tr. Mosk. Mat. Obshch., 10, 297–350 (1961).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
W. Walter, “Differential inequalities,” Lect. Notes Pure Appl. Math., 129, 249–283 (1991).
W. Walter, “Differential inequalities and maximum principles; theory, new methods and applications,” Nonlin. Anal. Appl., 30, 4695–4711 (1997).
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Struk, O.O., Tkachenko, V.I. On Impulsive Lotka–Volterra Systems with Diffusion. Ukrainian Mathematical Journal 54, 629–646 (2002). https://doi.org/10.1023/A:1021039528818
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DOI: https://doi.org/10.1023/A:1021039528818