×

On the periodic solutions of periodic multi-species competitive systems with delays. (English) Zbl 1035.34078

By means of Schauder’s fixed-point theorem, the author investigates a general n-species periodic multi-species competition system of Kolmogorov type with finite or infinite delays and derives sufficient conditions for the existence of positive periodic solutions. The result is applied to the single species logistic model and to an \(n\)-species Lotka-Volterra competition system.

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Freedman, H. I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 23, 3, 689 (1992) · Zbl 0764.92016
[2] Wang, W.; Chen, L.; Lu, Z., Global stability of a competition model with periodic coefficients and time delays, Can. Appl. Math. Quart., 3, 365 (1995) · Zbl 0845.92020
[3] Ahlip, R. A.; King, R. R., Global asymptotic stability of a periodic system of delay logistic equations, Bull. Aust. Math. Soc., 53, 373 (1996) · Zbl 0888.34061
[4] Gopalsamy, K., Global asymptotic stability in a periodic integrodifferential system, Tohoku Math. J., 37, 323 (1985) · Zbl 0587.45013
[5] Ahmad, S.; Mohana Rao, M. R., Asymptotically periodic solutions of \(n\)-competing species problem with time delays, J. Math. Anal. Appl., 186, 559 (1994) · Zbl 0818.45004
[6] Tang, B.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems, Tohoku Math. J., 49, 217 (1997) · Zbl 0883.34074
[7] Gopalsamy, K., Stability and Oscillations in Delay Differential Equation of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039
[8] Kuang, Y., Delay Differential Equation with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston, MA
[9] Smith, H. L., Monotone semiflows generated by functional differential equations, J. Different. Eq., 66, 420 (1987) · Zbl 0612.34067
[10] Smith, H. L.; Thieme, H. R., Strongly order preserving semiflows generated by functional differential equations, J. Different. Eq., 93, 332 (1991) · Zbl 0735.34065
[11] Horn, W. A., Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc., 149, 391 (1970) · Zbl 0201.46203
[12] Li, S.; Wen, L., Theory of Functional Differential Equations (1987), Science and Technology Press: Science and Technology Press Hunan, Changsha
[13] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (1985), Academic Press: Academic Press San Diego, CA · Zbl 0635.34001
[14] Deimling, K., Nonlinear Functional Anal. (1985), Springer: Springer Berlin
[15] Teng, Z., The almost periodic Kolmogorov competitive systems, Nonlinear Anal., 42, 1221 (2000) · Zbl 1135.34319
[16] Bereketoglu, H.; Goyri, I., Global asymptotical stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210, 279 (1997) · Zbl 0880.34072
[17] Ahmad, S.; Lazer, A. C., On the nonautonomous \(n\)-competing species problems, Applicable Anal., 57, 309 (1995) · Zbl 0859.34033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.