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Periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay. (English) Zbl 1003.34060

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution to a two-species nonautonomous competition Lotka-Volterra patch system with time delay \[ \begin{aligned} x'_1(t)&=x_1(t)[r_1(t)-a_1(t)x_1(t)-b_1(t)y(t)]+D_1(t)[x_2(t)-x_1(t)],\cr x'_2(t)&=x_2(t)[r_2(t)-a_2(t)x_2(t)]+D_2(t)[x_1(t)-x_2(t)],\cr y'(t)&=y(t) \Biggl[r_3(t)-a_3(t)x_1(t)-b_3(t)y(t)-\beta(t)\int_{-\tau}^{0}K(s)y(t+s) ds\Biggr], \end{aligned} \] is established, where \(r_i(t), a_i(t)\), \(i=1,2,3\), \(D_i(t)\), \(i=1,2\), \(b_i(t)\) \(i=1,3\), and \(\beta(t)\) is any positive periodic continuous function with period \(\omega>0\), \(\tau\) is a nonnegative constant and \(K(s)\) is a continuous nonnegative function on \([-\tau,0]\).

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
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References:

[1] Zhang, J. G.; Chen, L. S.; Chen, X. D., Persistence and global stability for two-species nonautonomous competition Lotka-Volterra patch-system with time delay, Nonlinear Anal., 37, 1019-1028 (1999) · Zbl 0949.34060
[2] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[3] Li, Y., Periodic solution of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057
[4] Li, Y., On a periodic neutral delay Lotka-Volterra system, Nonlinear Anal., 39, 767-778 (2000) · Zbl 0943.34058
[5] Ma, S. W.; Wang, Z. C.; Yu, J. S., Coincidence degree and periodic solutions of Duffing equations, Nonlinear Anal., 34, 443-460 (1998) · Zbl 0931.34048
[6] Ma, S. W.; Wang, Z. C.; Yu, J. S., An abstract existence theorem at resonance and its applications, J. Differential Equations, 145, 274-294 (1998) · Zbl 0940.34056
[7] Ma, S. W.; Wang, Z. C.; Yu, J. S., Periodic solutions of periodically perturbed functional-differential equations, Chinese Sci. Bull., 43, 1956-1959 (1998) · Zbl 0992.34054
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