Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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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  • [1] B. S. Goh, Stability in models of mutualism, American Naturalist 113, 261-275 (1979) MR 596837
  • [2] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math. 46, 368-375 (1986) MR 841454
  • [3] H.-I. Freedman and H. L. Smith, Tridiagonal competitive-cooperative Kolmogorov systems, Differential Equations and Dynamical Systems (submitted) MR 1386755
  • [4] G. Karakostas and I. Györi, Global stability in job systems, J. Math. Anal. Appl. 131, 85-96 (1988) MR 934432
  • [5] J. R. Gantmacher, The Theory of Matrices, Vol. II, Chelsea, New York, 1964
  • [6] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965 MR 0190463
  • [7] M. W. Hirsch, Systems of differential equations which are competitive or cooperative I. Limit sets, SIAM J. Math. Anal. 13, 167-179 (1982) MR 647119
  • [8] M. W. Hirsch, Systems of differential equations that are competitive or cooperative II. Convergence almost everywhere, SIAM J. Math. Anal. 16, 423-439 (1985) MR 783970
  • [9] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. reine angew. Math. 383, 1-53 (1988) MR 921986
  • [10] J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38, 80-103 (1980) MR 592869
  • [11] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995 MR 1315301
  • [12] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math. 46, 856-873 (1986) MR 858998

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Additional Information

DOI: https://doi.org/10.1090/qam/1604872
Article copyright: © Copyright 1998 American Mathematical Society

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