On the finite-dimensional dynamical systems with limited competition

Authors:
Xing Liang and Jifa Jiang

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3535-3554

MSC (2000):
Primary 34D23, 47H07; Secondary 92B05

DOI:
https://doi.org/10.1090/S0002-9947-02-03032-5

Published electronically:
April 30, 2002

MathSciNet review:
1911510

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Abstract: The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.

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Additional Information

**Xing Liang**

Affiliation:
Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, P. R. China

Email:
xliang@mail.ustc.edu.cn

**Jifa Jiang**

Affiliation:
Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, P. R. China

Email:
jiangjf@ustc.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-02-03032-5

Keywords:
Map with limited competition,
index of fixed points,
global stability,
permanence,
coexistence

Received by editor(s):
May 25, 2001

Published electronically:
April 30, 2002

Additional Notes:
Research supported by the National Natural Science Foundation of China

Article copyright:
© Copyright 2002
American Mathematical Society