×

Existence and stability of almost periodic solutions of nonautonomous competitive systems with weak Allee effect and delays. (English) Zbl 1221.34185

Summary: A class of nonautonomous Lotka–Volterra type multispecies competitive systems with weak Allee effect and delays are considered. By using Mawhin’s continuation theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of almost periodic solutions for the Lotka–Volterra system. On the case of no delays of Allee effects, by constructing a suitable Lyapunov function, we get a sufficient condition for the globally attractivity of the almost periodic solution for the Lotka–Volterra system. Moreover, we also present an illustrative example to show the effectiveness of our results.

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fang, H.; Xiao, Y., Existence of multiple periodic solutions for delay Lotka-Volterra competitive patch systems with harvesting, Math Comput Model, 33, 1086-1096 (2009) · Zbl 1168.34349
[2] Li, Y.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J Math Anal Appl, 255, 260-280 (2001) · Zbl 1024.34062
[3] Lin, W.; Chen, T., Positive periodic solutions of delayed periodic Lotka-Volterra systems, Phys Lett A, 334, 273-287 (2005) · Zbl 1123.34330
[4] Li, Y., Positive periodic solutions of discrete Lotka-Volterra competition systems with state dependent and distributed delays, Appl Math Comput, 190, 526-531 (2007) · Zbl 1125.39006
[5] Li, Y., Positive periodic solutions of periodic neutral Lotka-Volterra system with state dependent delays, J Math Anal Appl, 330, 1347-1362 (2007) · Zbl 1118.34059
[6] Xia, Y.; Cao, J.; Cheng, S., Positive solutions for a Lotka-Volterra mutualism system with several delays, Appl Math Model, 31, 1960-1969 (2007) · Zbl 1167.34343
[7] Li, Y., Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays, Chaos Solitons Fract, 37, 288-298 (2008) · Zbl 1145.34361
[8] Li Y, Fan X. Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients. Appl Math Model, in press. doi:10.1016/j.apm.2008.05.013; Li Y, Fan X. Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients. Appl Math Model, in press. doi:10.1016/j.apm.2008.05.013 · Zbl 1205.34086
[9] Yu, Y.; Cai, M., Existence and exponential stability of almost-periodic solutions for higher-order Hopfield neural networks, Math Comput Model, 47, 943-951 (2008) · Zbl 1144.34370
[10] Chen, L.; Zhao, H., Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients, Chaos Solitons Fract, 35, 351-357 (2008) · Zbl 1140.34425
[11] Meng, X.; Jiao, J.; Chen, L., Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays, Nonlinear Anal, 68, 3633-3645 (2008) · Zbl 1155.34042
[12] Qi, W.; Dai, B., Almost periodic solution for Lotka-Volterra competitive system with delay and feedback controls, Appl Math Comput, 200, 133-146 (2008) · Zbl 1146.93021
[13] Meng, X.; Chen, L., Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays, J Theor Biol, 243, 562-574 (2006) · Zbl 1447.92355
[14] Fink, A., Almost periodic differential equations, (Lecture notes in mathematics, vol. 377 (1974), Springer: Springer Berlin) · Zbl 0325.34039
[15] Ezzinbi, K.; Hachimi, M. A., Existence of positive almost periodic solutions of functional equations via Hilbert’s projective metric, Nonlinear Anal, 26, 6, 1169-1176 (1996) · Zbl 0857.45005
[16] Gaines, R.; Mawhin, J., Coincidence degree and nonlinear differential equations (1977), Springer Verlag: Springer Verlag Berlin · Zbl 0326.34021
[17] Shi, Junping; Shivaji, Ratnasingham, Persistence in reaction diffusion models with weak Allee effect, J Math Biol, 52, 6, 807-829 (2006) · Zbl 1110.92055
[18] Thieme, Horst R., Mathematics in population biology, (Princeton series in theoretical and computational biology (2003), Princeton University Press: Princeton University Press Princeton (NJ)) · Zbl 1054.92042
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.