Persistence definitions and their connections

Freedman, H. I.; Moson, P.

doi:10.1090/S0002-9939-1990-1012928-6

Persistence definitions and their connections
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by H. I. Freedman and P. Moson PDF

Proc. Amer. Math. Soc. 109 (1990), 1025-1033 Request permission

Abstract:

We give various definitions of types of persistence of a dynamical system and establish a hierarchy among them by proving implications and demonstrating counterexamples. Under appropriate conditions, we show that several of the definitions are equivalent.

References

T. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137 (1989), no. 1, 240–263. MR 981936, DOI 10.1016/0022-247X(89)90287-4
Geoffrey Butler, H. I. Freedman, and Paul Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), no. 3, 425–430. MR 822433, DOI 10.1090/S0002-9939-1986-0822433-4
Geoffrey Butler and Paul Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), no. 2, 255–263. MR 848269, DOI 10.1016/0022-0396(86)90049-5
Alessandro Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc. 104 (1988), no. 1, 111–116. MR 958053, DOI 10.1090/S0002-9939-1988-0958053-2
H. I. Freedman and Paul Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci. 68 (1984), no. 2, 213–231. MR 738903, DOI 10.1016/0025-5564(84)90032-4
H. I. Freedman and Paul Waltman, Persistence in a model of three competitive populations, Math. Biosci. 73 (1985), no. 1, 89–101. MR 779763, DOI 10.1016/0025-5564(85)90078-1
Thomas C. Gard, Uniform persistence in multispecies population models, Math. Biosci. 85 (1987), no. 1, 93–104. MR 904450, DOI 10.1016/0025-5564(87)90101-5
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Jack K. Hale and Paul Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), no. 2, 388–395. MR 982666, DOI 10.1137/0520025
J. Hofbauer and K. Sigmund, Permanence for replicator equations, Dynamical systems (Sopron, 1985) Lecture Notes in Econom. and Math. Systems, vol. 287, Springer, Berlin, 1987, pp. 70–91. MR 1120043, DOI 10.1007/978-3-662-00748-8_{7}
V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), no. 4, 267–275. MR 776353, DOI 10.1007/BF01540776
V. Hutson and R. Law, Permanent coexistence in general models of three interacting species, J. Math. Biol. 21 (1985), no. 3, 285–298. MR 804152, DOI 10.1007/BF00276227

Permanence in dynamical systems

Gabriela Kirlinger, Permanence in Lotka-Volterra equations: linked prey-predator systems, Math. Biosci. 82 (1986), no. 2, 165–191. MR 871637, DOI 10.1016/0025-5564(86)90136-7
Robert M. May and Warren J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), no. 2, 243–253. MR 392035, DOI 10.1137/0129022
V. A. Pliss, Nonlocal problems of the theory of oscillations, Academic Press, New York-London, 1966. Translated from the Russian by Scripta Technica, Inc; Translation edited by Harry Herman. MR 0196199

Stability theory by Liapunov’s direct method

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P. Schuster, K. Sigmund, and R. Wolff, On $\omega$-limits for competition between three species, SIAM J. Appl. Math. 37 (1979), no. 1, 49–54. MR 536302, DOI 10.1137/0137004
Josef Hofbauer and Joseph W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1137–1142. MR 984816, DOI 10.1090/S0002-9939-1989-0984816-4