Abstract
The authors study the ω-limit set dichotomy of the Kolmogorov systems \(\dot x\) i=xi f i(x)x i≥0, 1≤i≤n with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Coppel, W.A.: Stability and Asymptotic Behavior of Ordinary Differential Equations. D. C. Health, Boston, 1965.
Hartman, P.: Ordinary Differential Equations, Boston, 1982.
Hirsch, M.W.: Systems of differential equations which are competitive or cooperative. I: Limit Sets. SIAM J. Math. Anal., 13 (1982), 161–197.
Hirsch, M.W.: Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal., 16 (1985), 423–439.
Hirsch, M.W.: Stability and convergence in strongly monotone dynamical systems, J. reine angew. Math. 383 (1988), 1–53.
Hirsch, M.W.: Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity 1 (1988), 51–71.
Hirsch, M.W. and Pugh, C.C.: Cohomology of chain recurrent sets, Ergod. Th. Dynam. Sys. 8 (1988), 73–80.
Jiang, J. and Tu, C.F.: The asymptotic behavior of models of cooperation with a concavity property. Nonlinear analysis: Real World Application 1(2000) 329–344.
Smith, H.L.: Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jiang, J., Wang, Y. On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems. Positivity 7, 185–194 (2003). https://doi.org/10.1023/A:1026287609893
Issue Date:
DOI: https://doi.org/10.1023/A:1026287609893