The complete classification of asymptotic behavior for bounded cooperative Lotka-Volterra systems with the assumption (SM)
Author:
J. F. Jiang
Journal:
Quart. Appl. Math. 56 (1998), 37-53
MSC:
Primary 92D25; Secondary 34C11
DOI:
https://doi.org/10.1090/qam/1604872
MathSciNet review:
MR1604872
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Abstract: Assume that the $n$-dimensional Lotka-Volterra system is cooperative with the assumption $\left ( SM \right ) : {a_{ii}} < 0, {a_{ij}} > 0$ for all $i \ne j$, and all solutions of the system are bounded. Then we give a complete classification of the behavior of the solutions. This classification does not require any restriction on the auto-increase coefficients.
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B. S. Goh, Stability in models of mutualism, American Naturalist 113, 261–275 (1979)
H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math. 46, 368–375 (1986)
H.-I. Freedman and H. L. Smith, Tridiagonal competitive-cooperative Kolmogorov systems, Differential Equations and Dynamical Systems (submitted)
G. Karakostas and I. Györi, Global stability in job systems, J. Math. Anal. Appl. 131, 85–96 (1988)
J. R. Gantmacher, The Theory of Matrices, Vol. II, Chelsea, New York, 1964
W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965
M. W. Hirsch, Systems of differential equations which are competitive or cooperative I. Limit sets, SIAM J. Math. Anal. 13, 167–179 (1982)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative II. Convergence almost everywhere, SIAM J. Math. Anal. 16, 423–439 (1985)
M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. reine angew. Math. 383, 1–53 (1988)
J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38, 80–103 (1980)
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995
H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math. 46, 856–873 (1986)
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© Copyright 1998
American Mathematical Society