×

Uniformly persistent systems. (English) Zbl 0603.34043

Conditions are given under which weak persistence of a dynamical system with respect to the boundary of a given set implies uniform persistence.

MSC:

37C10 Dynamics induced by flows and semiflows
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. P. Bhatia and G. P. Szegő, Dynamical systems: Stability theory and applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967. · Zbl 0155.42201
[2] Herbert I. Freedman, Deterministic mathematical models in population ecology, Monographs and Textbooks in Pure and Applied Mathematics, vol. 57, Marcel Dekker, Inc., New York, 1980. · Zbl 0448.92023
[3] H. I. Freedman and Paul Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci. 68 (1984), no. 2, 213 – 231. · Zbl 0534.92026
[4] H. I. Freedman and Paul Waltman, Persistence in a model of three competitive populations, Math. Biosci. 73 (1985), no. 1, 89 – 101. · Zbl 0584.92018
[5] Thomas C. Gard, Persistence in food chains with general interactions, Math. Biosci. 51 (1980), no. 1-2, 165 – 174. · Zbl 0453.92017
[6] Thomas C. Gard, Top predator persistence in differential equation models of food chains: the effects of omnivory and external forcing of lower trophic levels, J. Math. Biol. 14 (1982), no. 3, 285 – 299. · Zbl 0494.92022
[7] Thomas C. Gard and Thomas G. Hallam, Persistence in food webs. I. Lotka-Volterra food chains, Bull. Math. Biol. 41 (1979), no. 6, 877 – 891. · Zbl 0422.92017
[8] J. K. Hale and A. S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol. 18 (1983), no. 3, 255 – 280. · Zbl 0525.92024
[9] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133 – 163.
[10] Josef Hofbauer, A general cooperation theorem for hypercycles, Monatsh. Math. 91 (1981), no. 3, 233 – 240. · Zbl 0449.34039
[11] S. B. Hsu, S. P. Hubbell, and Paul Waltman, Competing predators, SIAM J. Appl. Math. 35 (1978), no. 4, 617 – 625. · Zbl 0394.92025
[12] V. Hutson and G. T. Vickers, A criterion for permanent coexistence of species, with an application to a two-prey one-predator system, Math. Biosci. 63 (1983), no. 2, 253 – 269. · Zbl 0524.92023
[13] Robert M. May and Warren J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), no. 2, 243 – 253. Special issue on mathematics and the social and biological sciences. · Zbl 0314.92008
[14] Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. · Zbl 0246.58012
[15] P. Schuster, K. Sigmund, and R. Wolff, On \?-limits for competition between three species, SIAM J. Appl. Math. 37 (1979), no. 1, 49 – 54. · Zbl 0418.92016
[16] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747 – 817. · Zbl 0202.55202
[17] Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. · Zbl 0061.39301
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.