×

Permanence for an integrodifferential model of mutualism. (English) Zbl 1118.45006

The authors obtain sufficient conditions, using the differential inequality theory, to ensure permanence of a two system of the integro-differential equation model for mutualism.

MSC:

45J05 Integro-ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Li, Y. K.; Xu, G. T., Positive periodic solutions for an integrodifferential model of mutualism, Appl. Math. Lett., 14, 525-530 (2001) · Zbl 0981.45002
[2] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0752.34039
[3] Dean, A. M., A simple model of mutualism, Am. Natural, 121, 409-417 (1983)
[4] Boucher, D. H., The Biology of Mutualism, Ecology and Evolution (1985), Croom Helm: Croom Helm London
[6] Cui, J. A., Global asymptotic stability in \(n\)-species cooperative system with time delays, Syst. Sci. Math. Sci., 7, 1, 45-48 (1994) · Zbl 0807.34088
[7] Yang, P.; Xu, R., Global asymptotic stability of periodic solution in \(n\)-species cooperative system with time delays, J. Biomath., 13, 6, 841-846 (1998)
[8] Zhang, X.; Wang, K., Almost periodic solution for \(n\)-species cooperative system with time delay, J. Northeast Normal Univ., 34, 3, 9-13 (2002)
[10] Zhao, J. D.; Jiang, J. F., Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system, J. Math. Anal. Appl., 299, 663-675 (2004) · Zbl 1066.34050
[11] Chen, F. D.; Xie, X. D.; Shi, J. L., Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math., 194, 2, 368-387 (2006) · Zbl 1104.34050
[13] Lisena, B., Competitive exclusion in a periodic Lotka-Volterra system, Appl. Math. Comput., 177, 2, 761-768 (2006) · Zbl 1100.92070
[14] Muroya, Y., Boundedness and partial survival of species in nonautonomous Lotka-Volterra systems, Nonlinear Anal.: Real World Appl., 6, 2, 263-272 (2005) · Zbl 1077.34077
[16] Tineo, A., Asymptotic behavior of positive solutions of the nonautonomous Lotka-Volterra competition equations, Differen. Integral Equat., 6, 419-457 (1993) · Zbl 0774.34037
[17] Hirsch, W.; Hanisch, H.; Gabriel, J., Differential equation models of some parasitic infection-methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38, 733-753 (1985) · Zbl 0637.92008
[18] Chen, F. D., Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model, Nonlinear Anal.: Real World Appl., 7, 4, 895-915 (2006) · Zbl 1119.34038
[19] Fan, M.; Wong, Patrica J. Y.; Agarwal, Ravi P., Periodicity and stability in periodic \(n\)-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica, 19, 4, 801-822 (2003) · Zbl 1047.34080
[20] Chen, F. D.; Chen, X. X.; Chen, A. P.; Cao, J. D., Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control, Acta Math. Sinica, 21, 6, 1319-1336 (2005) · Zbl 1110.34049
[21] Li, W. T.; Wang, L. L., Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control, J. Comput Appl. Math., 180, 2, 293-309 (2005) · Zbl 1069.34100
[22] Chen, F. D., Global asymptotic stability in \(n\)-species nonautonomous Lotka-Volterra competitive systems with infinite delays and feedback control, Appl. Math. Comput., 170, 2, 1452-1468 (2005) · Zbl 1081.92038
[23] Chen, F. D., The permanence and global attractivity of Lotka-Volterra competition system with feedback controls, Nonlinear Anal.: Real World Appl., 7, 1, 133-143 (2006) · Zbl 1103.34038
[24] Chen, F. D.; Lin, F. X.; Chen, X. X., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158, 1, 45-68 (2004) · Zbl 1096.93017
[25] Chen, F. D., Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162, 3, 1279-1302 (2005) · Zbl 1125.93031
[26] Chen, F. D., On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180, 1, 33-49 (2005) · Zbl 1061.92058
[28] Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0425.34048
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.