Persistence definitions and their connections
Authors:
H. I. Freedman and P. Moson
Journal:
Proc. Amer. Math. Soc. 109 (1990), 10251033
MSC:
Primary 34C35; Secondary 54H20, 58F25, 92A15
MathSciNet review:
1012928
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [1]
T. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137 (1989), 240263. MR 981936 (90d:58141)
 [2]
G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), 425429. MR 822433 (87d:58119)
 [3]
, Persistence in dynamical systems, J. Differential Equations 63 (1986), 255263. MR 848269 (87k:54058)
 [4]
A. Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc. 104 (1988), 111116. MR 958053 (90a:34094)
 [5]
H. I. Freedman and P. Waltman, Persistence in models of three interacting predatorprey populations, Math. Biosci. 68 (1984), 213231. MR 738903 (85h:92037)
 [6]
, Persistence in a model of three competitive populations, Math. Biosci. 73 (1985), 89101. MR 779763 (86i:92038)
 [7]
T. C. Gard, Uniform persistence in multispecies population models, Math. Biosci. 85 (1987), 93104. MR 904450 (88m:92040)
 [8]
J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Applied Math. Sci. 42 (1983). MR 709768 (85f:58002)
 [9]
J. K. Hale and P. Waltman, Persistence in infinitedimensional systems, SIAM J. Math. Anal. 20 (1989), 388395. MR 982666 (90b:58156)
 [10]
J. Hofbauer and K. Sigmund, Permanence for replicator equations, Lecture Notes in Economics and Math. Systems, vol. 287, Springer (1987), 7085. MR 1120043
 [11]
V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), 267275. MR 776353 (86c:34086)
 [12]
V. Hutson and R. Law, Permanent coexistence in general models of three interacting species, J. Math. Biol. 21 (1985), 285298. MR 804152 (87b:92026)
 [13]
V. Hutson and K. Schmitt, Permanence in dynamical systems, preprint.
 [14]
G. Kirlinger, Permanence in LotkaVolterra equations: Linked predatorprey systems, Math. Biosci. 82 (1986), 165191. MR 871637 (88a:92043)
 [15]
R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), 243253. MR 0392035 (52:12853)
 [16]
V. A. Pliss, Nonlocal problems of the theory of oscillations, Academic Press, New York, 1966. MR 0196199 (33:4391)
 [17]
H. Rouche, P. Habets and M. Laloy, Stability theory by Liapunov's direct method, Appl. Math. Sci. 22 (1977).
 [18]
P. Schuster, K. Sigmund and R. Wolff, On limits for competition between three species, SIAM J. Appl. Math. 37 (1979), 4954. MR 536302 (80g:92029)
 [19]
J. Hofbauer and J. WH. So, Uniform persistence and repellers for maps, Proc. Amer. Math. Soc. 107 (1989), 11371142. MR 984816 (90i:58096)
 [1]
 T. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137 (1989), 240263. MR 981936 (90d:58141)
 [2]
 G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), 425429. MR 822433 (87d:58119)
 [3]
 , Persistence in dynamical systems, J. Differential Equations 63 (1986), 255263. MR 848269 (87k:54058)
 [4]
 A. Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc. 104 (1988), 111116. MR 958053 (90a:34094)
 [5]
 H. I. Freedman and P. Waltman, Persistence in models of three interacting predatorprey populations, Math. Biosci. 68 (1984), 213231. MR 738903 (85h:92037)
 [6]
 , Persistence in a model of three competitive populations, Math. Biosci. 73 (1985), 89101. MR 779763 (86i:92038)
 [7]
 T. C. Gard, Uniform persistence in multispecies population models, Math. Biosci. 85 (1987), 93104. MR 904450 (88m:92040)
 [8]
 J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Applied Math. Sci. 42 (1983). MR 709768 (85f:58002)
 [9]
 J. K. Hale and P. Waltman, Persistence in infinitedimensional systems, SIAM J. Math. Anal. 20 (1989), 388395. MR 982666 (90b:58156)
 [10]
 J. Hofbauer and K. Sigmund, Permanence for replicator equations, Lecture Notes in Economics and Math. Systems, vol. 287, Springer (1987), 7085. MR 1120043
 [11]
 V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), 267275. MR 776353 (86c:34086)
 [12]
 V. Hutson and R. Law, Permanent coexistence in general models of three interacting species, J. Math. Biol. 21 (1985), 285298. MR 804152 (87b:92026)
 [13]
 V. Hutson and K. Schmitt, Permanence in dynamical systems, preprint.
 [14]
 G. Kirlinger, Permanence in LotkaVolterra equations: Linked predatorprey systems, Math. Biosci. 82 (1986), 165191. MR 871637 (88a:92043)
 [15]
 R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), 243253. MR 0392035 (52:12853)
 [16]
 V. A. Pliss, Nonlocal problems of the theory of oscillations, Academic Press, New York, 1966. MR 0196199 (33:4391)
 [17]
 H. Rouche, P. Habets and M. Laloy, Stability theory by Liapunov's direct method, Appl. Math. Sci. 22 (1977).
 [18]
 P. Schuster, K. Sigmund and R. Wolff, On limits for competition between three species, SIAM J. Appl. Math. 37 (1979), 4954. MR 536302 (80g:92029)
 [19]
 J. Hofbauer and J. WH. So, Uniform persistence and repellers for maps, Proc. Amer. Math. Soc. 107 (1989), 11371142. MR 984816 (90i:58096)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
34C35,
54H20,
58F25,
92A15
Retrieve articles in all journals
with MSC:
34C35,
54H20,
58F25,
92A15
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010129286
PII:
S 00029939(1990)10129286
Article copyright:
© Copyright 1990
American Mathematical Society
