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Uniform persistence and flows near a closed positively invariant set. (English) Zbl 0811.34033
The behavior of a continuous flow in the vicinity of a closed positively invariant set in a metric space is studied. The obtained results generalize results of Ura-Kimura and Bhatia on classification of a flow near a compact invariant set in a locally compact metric space. Applying the obtained results, the authors prove two persistence theorems. One of the theorems unifies and generalizes earlier persistence results based on Lyapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes persistence results based on analysis of a flow on the boundary by relaxing point dissipativity and invariance of the boundary. The obtained results are illustrated by considering several ecological systems.

37-99Dynamic systems and ergodic theory (MSC2000)
37C10Vector fields, flows, ordinary differential equations
34D05Asymptotic stability of ODE
34G20Nonlinear ODE in abstract spaces
Full Text: DOI
[1] Bhatia, N. P. (1969). Dynamical systems. In Kuhn, H. W., and Szegö, G. P. (eds.),Mathematical Systems Theory and Economics, Springer-Verlag, New York, pp. 1--10. · Zbl 0183.37201
[2] Bhatia, N. P. (1970). Attraction and non-saddle sets in dynamical systems.J. Diff. Eqs. 8, 229--249. · Zbl 0207.08803 · doi:10.1016/0022-0396(70)90003-3
[3] Bhatia, N. P., and Szegö, G. P. (1970).Stability Theory of Dynamical Systems, Springer-Verlag, Berlin. · Zbl 0213.10904
[4] Butler, G. J., and Waltman, P. (1986). Persistence in dynamical systems.J. Diff. Eqs. 63, 255--263. · Zbl 0603.58033 · doi:10.1016/0022-0396(86)90049-5
[5] Butler, G. J., Freedman, H. I., and Waltman, P. (1986). Uniformly persistent systems.Proc. Am. Math. Soc. 96, 425--430. · Zbl 0603.34043 · doi:10.1090/S0002-9939-1986-0822433-4
[6] Conley, C. (1978).Isolated Invariant Sets and the Morse Index, CBMS, Vol. 38, Providence, RI. · Zbl 0397.34056
[7] Dunbar, S. R., Rybakowski, K. P., and Schmitt, K. (1986). Persistence in models of predatorprey populations with diffusion.J. Diff. Eqs. 65, 117--138. · Zbl 0605.34044 · doi:10.1016/0022-0396(86)90044-6
[8] Fernandes, M., and Zanolin, F. (1990). Repelling conditions for boundary sets using Liapunov-like functions. II. Persistence and periodic solutions.J. Diff. Eqs. 86, 33--58. · Zbl 0719.34092 · doi:10.1016/0022-0396(90)90039-R
[9] Fonda, A. (1988). Uniformly persistent semidynamical systems.Proc. Am. Math. Soc. 104, 111--116. · Zbl 0667.34065 · doi:10.1090/S0002-9939-1988-0958053-2
[10] Freedman, H. I., and Moson, P. (1990). Persistence definitions and their connections.Proc. Am. Math. Soc. 109, 1025--1033. · Zbl 0695.34049 · doi:10.1090/S0002-9939-1990-1012928-6
[11] Freedman, H. I., and Ruan, S. (1994). Uniform persistence in functional differential equations.J. Diff. Eqs. (in press). · Zbl 0814.34064
[12] Freedman, H. I., and So, J. (1989). Persistence in discrete semidynamical systems.SIAM J. Math. Anal. 20, 930--938. · Zbl 0676.92011 · doi:10.1137/0520062
[13] Freedman, H. I., and Waltman, P. (1984). Persistence in models of three interacting predatorprey populations.Math. Biosci. 68, 213--231. · Zbl 0534.92026 · doi:10.1016/0025-5564(84)90032-4
[14] Garay, B. M. (1989). Uniform persistence and chain recurrence.J. Math. Anal. Appl. 139, 372--381. · Zbl 0677.54033 · doi:10.1016/0022-247X(89)90114-5
[15] Gard, T. C. (1987). Uniform persistence in multispecies population models.Math. Biosci. 85, 93--104. · Zbl 0631.92012 · doi:10.1016/0025-5564(87)90101-5
[16] Gard, T. C., and Hallam, T. G. (1979). Persistence of food webs. I. Lotka-Volterra food chains.Bull. Math. Biol. 41, 877--891. · Zbl 0422.92017
[17] Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI. · Zbl 0642.58013
[18] Hale, J. K., and Waltman, P. (1989). Persistence in infinite dimensional systems.SIAM J. Math. Anal. 20, 388--395. · Zbl 0692.34053 · doi:10.1137/0520025
[19] Hethcote, H. W., and van den Driessche, P. (1991). Some epidemiological models with nonlinear incidence.J. Math. Biol. 29, 271--287. · Zbl 0722.92015 · doi:10.1007/BF00160539
[20] Hofbauer, J. (1981). A general cooperation theorem for hypercycles.Monatsh. Math. 91, 233--240. · Zbl 0449.34039 · doi:10.1007/BF01301790
[21] Hofbauer, J. (1989). A unified approach to persistence.Acta Appl. Math. 14, 11--22. · Zbl 0669.92020 · doi:10.1007/BF00046670
[22] Hofbauer, J., and Sigmund, K. (1988).Dynamical Systems and the Theory of Evolution, Cambridge University Press, Cambridge. · Zbl 0678.92010
[23] Hofbauer, J., and So, J. (1989). Uniform persistence and repellers for maps.Proc. Am. Math. Soc. 107, 1137--1142. · Zbl 0678.58024 · doi:10.1090/S0002-9939-1989-0984816-4
[24] Hutson, V. (1984). A theorem on average Liapunov functions.Monatsh. Math. 98, 267--275. · Zbl 0542.34043 · doi:10.1007/BF01540776
[25] Hutson, V., and Schmitt, K. (1992). Permanence and the dynamics of biological systems.Math. Biosci. 111, 1--71. · Zbl 0783.92002 · doi:10.1016/0025-5564(92)90078-B
[26] Liu, W. M., Hethcote, H. W., and Levin, S. A. (1987). Dynamical behavior of epidemiological models with nonlinear incidence rates.J. Math. Biol. 25, 359--380. · Zbl 0621.92014 · doi:10.1007/BF00277162
[27] Sell, G. R. (1967). Nonautonomous differential equations and topological dynamics. I. The basic theory.Trans. Am. Math. Soc. 127, 247--262. · Zbl 0189.39602
[28] Sell, G. R., and Sibuya, Y. (1967). Behavior of solutions near a critical point. In Harris, W. A., Jr., and Sibuya, Y. (eds.),Proceedings United States-Japan Seminar on Differential and Functional Equations, Benjamin, New York, pp. 501--506. · Zbl 0189.38404
[29] Tang, M. (1990). Persistence in a higher dimensional population dynamical systems.Acta Math. Appl. Sinica 13, 431--443. · Zbl 0729.92027
[30] Teng, Z.-D., and Duan, K.-C. (1990). Persistence in dynamical systems.Q. Appl. Math. 48, 463--472 · Zbl 0722.92017
[31] Thieme, H. R. (1993). Persistence under relaxed point-dissipativity (with application to an endemic model).SIAM J. Math. Anal. 24, 407--435. · Zbl 0774.34030 · doi:10.1137/0524026
[32] Ura, T., and Kimura, I. (1960). Sur le courant extérieur à une région invariante: Théorème de Bendixson.Comm. Math. Univ. St. Paul 8, 23--39.
[33] Waltman, P. (1992). A brief survey of persistence in dynamical systems. In Busenberg, S., and Martelli, M. (eds.),Delay Differential Equations and Dynamical Systems, Springer-Verlag, New York, pp. 31--40. · Zbl 0756.34054
[34] Whyburn, G. T. (1942).Analytic Topology, Am. Math. Soc. Colloq. Publ. Vol. 28, Providence, RI. · Zbl 0061.39301
[35] Yang, F., and Ruan, S. (1992). A generalization of the Butler-McGehee lemma and its applications in persistence theory. Preprint · Zbl 0879.34047