zbMATH — the first resource for mathematics

Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model. (English) Zbl 0954.92027
Summary: We investigate a generalized \(n\)-species Gilpin-Ayala competition system [M.E. Gilpin and F.J. Ayala, Proc. Natl. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)] with several deviating arguments in periodic environment, which is more general and more realistic than the classical Lotka-Volterra competition systems. By using the method of coincidence degree, a set of easily verifiable sufficient conditions is derived for the existence of at least one strictly positive (componentwise) periodic solution.
Some new results are obtained. As applications, we also apply our main results to some special cases of the system we consider here, including the classical \(n\)-species Lotka-Volterra competition systems and \(n\)-species Gilpin-Ayala competition model, which have been studied extensively in the literature. Some known results are improved and generalized. The examples show that our criteria are new, general, and easily verifiable.

92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
Full Text: DOI
[1] Ahmad, S., On the non autonomous Volterra-Lotka competition equations, (), 199-204 · Zbl 0848.34033
[2] Alvarez, C.; Lazer, A.C., An application of topological degree to the periodic competing species model, J. austral. math. soc., 28, 202-219, (1986), Ser. B · Zbl 0625.92018
[3] Battaaz, A.; Zanolin, F., Coexistence states for periodic competitive Kolmogorov systems, J. math. anal. appl., 219, 179-199, (1998) · Zbl 0911.34037
[4] Cushing, J.M., Two species competition in a periodic environment, J. math. biol., 10, 385-400, (1980) · Zbl 0455.92012
[5] Fan, M.; Wang, K., Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition system with several deviating arguments, Math. biosci., 160, 47-61, (1999) · Zbl 0964.34059
[6] Golpalsamy, K., Globally asymptotic stability in a periodic Lotka-Volterra system, J. austral. math. soc. ser. B, 24, 160-170, (1982)
[7] Golpalsamy, K., Globally asymptotic stability in a periodic Lotka-Volterra system, J. austral. math. soc. ser. B, 29, 66-72, (1985) · Zbl 0588.92019
[8] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics, mathematics and its applications 74, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[9] Hale, J.K.; Somolinos, A.S., Competition for fluctuating nutrient, J. math. biol., 18, 225-280, (1983) · Zbl 0525.92024
[10] Korman, P., Some new results on the periodic competition model, J. math. anal. appl., 171, 131-138, (1992) · Zbl 0848.34026
[11] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press Boston, MA
[12] Li, Y.K., Periodic solutions of N-species competition system with time delays, J. biomathematics, 12, 1, 1-7, (1997) · Zbl 0891.92027
[13] May, R.M.; Leonard, W.J., Nonlinear aspects of competition between three species, SIAM J. appl. math., 29, 243-253, (1975) · Zbl 0314.92008
[14] de Mottoni, P.; Schiaffino, A., Competition systems with periodic coefficients: A geometric approach, J. math. biol., 11, 319-335, (1981) · Zbl 0474.92015
[15] Shibata, A.; Saito, N., Time delays and chaos in two competition system, Math. biosci., 51, 199-211, (1980) · Zbl 0455.92011
[16] Smith, H.L., Periodic solutions of periodic competitive and cooperative systems, SIAM J. math. anal., 17, 1289-1318, (1986) · Zbl 0609.34048
[17] Smith, H.L., Periodic competitive differential equations and the discrete dynamics of a competitive map, J. diff. eqns., 64, 165-194, (1986) · Zbl 0596.34013
[18] Tineo, A.; Alvarez, C., A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J. math. anal. appl., 159, 44-50, (1991) · Zbl 0729.92025
[19] Gilpin, M.E.; Ayala, F.J., Global models of growth and competition, (), 3590-3593 · Zbl 0272.92016
[20] Ayala, F.J.; Gilpin, M.E.; Eherenfeld, J.G., Competition between species: theoretical models and experimental tests, Theoretical population biology, 4, 331-356, (1973)
[21] Gilpin, M.E.; Ayala, F.J., Schoener’s model and drosophila competition, Theoretical population biology, 9, 12-14, (1976)
[22] Thomas, W.R.; Powerantz, M.J.; Gilpin, M.E., Chaos, asymmetric growth and group selection for dynamical stability, Ecology, 61, 1312-1320, (1980)
[23] Gilpin, M.E.; Carpenter, M.P.; Powerantz, M.J., The assembly laboratory community: multi species competition in drosophila, ()
[24] Goh, B.S.; Agnew, T.T., Stability in gilpin and Ayala’s model of competition, J. math. biol., 4, 275-279, (1977) · Zbl 0379.92017
[25] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific Netherlands · Zbl 0453.92015
[26] Liao, X.X.; Li, J., Stability in gilpin-ayala competition models with diffusion, Nonlinear analysis TMA, 28, 1751-1758, (1997) · Zbl 0872.35054
[27] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
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.