Ahmad, Shair; Lazer, Alan C. Average growth and total permanence in a competitive Lotka-Volterra System. (English) Zbl 1162.34329 Ann. Mat. Pura Appl. (4) 185, Suppl. 5, S47-S67 (2006). Summary: We consider a class of non-autonomous competitive Lotka-Volterra Systems. Such a system is called strongly permanent if small perturbations of the system are permanent. We define such a system to be totally permanent if the system as well as its subsystems are strongly permanent. When the growth rates have averages and the interaction coefficients are non-negative constants, we give a computable necessary and sufficient condition for the system to be totally permanent. Cited in 1 ReviewCited in 22 Documents MSC: 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general) Keywords:strongly permanent; strongly persistent; totally permanent; growth rate; interaction coefficients; exponential decay PDF BibTeX XML Cite \textit{S. Ahmad} and \textit{A. C. Lazer}, Ann. Mat. Pura Appl. (4) 185, S47--S67 (2006; Zbl 1162.34329) Full Text: DOI OpenURL References: [1] Ahmad, S., Lazer, A.C.: On Persistence and Extinction of Species, Centro Internacional de Matematica, vol. 20, pp. 1-12. Lisbon, Portugal 2002 [2] Ahmad, Nonlinear Anal., 34, 191 (1998) · Zbl 0934.34037 [3] Ahmad, Nonlinear Anal., 40, 37 (2000) · Zbl 0955.34041 [4] Ahmad, S., Lazer, A.C.: One species extinction in an autonomous competition model. Proc. of the First World Congress of Nonlinear Anal. Tampa, Florida, August 19-26, pp. 359-368. 1992 · Zbl 0846.34043 [5] Burton, Differ. Integral Equ., 4, 1269 (1991) [6] Butler, Proc. Am. Math. Soc., 96, 425 (1986) [7] Coste, SIAM J. Appl. Math., 36, 516 (1979) · Zbl 0412.92015 [8] Hale, SIAM J. Math. Anal., 20, 388 (1989) · Zbl 0692.34053 [9] Hallam, Math. Biosci., 46, 117 (1979) · Zbl 0413.92013 [10] Hofbauer, J., Sigmund, K.: The theory of evolution and dynamical systems. New York: Cambridge Univ. Press 1988 · Zbl 0678.92010 [11] Hutson, Math. Biosci., 111, 1 (1992) · Zbl 0783.92002 [12] Jansen, J. Math. Biol., 25, 411 (1986) · Zbl 0647.92021 [13] May, SIAM J. Appl. Math., 29, 243 (1975) · Zbl 0314.92008 [14] Mierczyński, J. Math. Anal. Appl., 267, 329 (2002) · Zbl 1017.34052 [15] Redheffer, Nonlinear Anal., 46, 1154 (2001) · Zbl 1003.34039 [16] Takeuchi, Y.: Global dynamical properties of Lotka-Volterra Systems. Singapore: World Scientific 1996 · Zbl 0844.34006 [17] Waltman, P.: A brief survey of persistence in dynamical systems. In: Delay Differential Equations and Dynamical Systems, ed. by S. Busenberg and M. Martelli. Berlin: Springer 1991 · Zbl 0756.34054 [18] Zanolin, Results Math., 21, 224 (1992) 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.