The DOI matches its Crossref record; 5 of the 5 references it cites were found in a registry or an index.
Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere
Open accessAbstract
A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone. Applications are made to generalizations of positive feedback loops.
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Cites
Differential Equations, Dynamical Systems, and Linear Algebra
(1974) · Pure and Applied Mathematics · doi:10.1016/s0079-8169(08)x6044-1 · Cited by 3,091
Stability and Asymptotic Behavior of Differential Equations.
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Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets
Hirsch, M. W. (1982) · SIAM Journal on Mathematical Analysis · doi:10.1137/0513013 · Cited by 443
Self-citationMathematics of cellular control processes II. Positive feedback to one gene
Griffith, J. S. (1968) · Journal of Theoretical Biology · doi:10.1016/0022-5193(68)90190-2 · Cited by 354
The struggle for life: I. Two species
Rescigno, A., & Richardson, I. W. (1967) · Bulletin of Mathematical Biology · doi:10.1007/bf02476909 · Cited by 101
Cited by
Nonlinear neural networks: Principles, mechanisms, and architectures
Grossberg, S. (1988) · Neural Networks · doi:10.1016/0893-6080(88)90021-4 · Cited by 1,623
Forti, M., & Tesi, A. (1995) · IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications · doi:10.1109/81.401145 · Cited by 745
Some new directions in control theory inspired by systems biology
Sontag, E. D. (2004) · Systems Biology · doi:10.1049/sb:20045006 · Cited by 171
Stable periodic solutions for the hypercycle system
Hofbauer, J., Mallet-Paret, J., & Smith, H. L. (1991) · Journal of Dynamics and Differential Equations · doi:10.1007/bf01049740 · Cited by 41
Metadata
- Authors
- Hirsch, M. W.
- Venue
- SIAM Journal on Mathematical Analysis
- Published
- 1985-05
- Volume
- 16
- Issue
- 3
- Pages
- 423-439
- ISSN
- 0036-1410 (ISSN-L)
- 1095-7154
- OpenAlex
- W2166127999
- Cited by
- 532
- References
- 5
- FWCI
- 14.21
- Type
- journal article
- Language
- en
- Open access
- green
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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