×

zbMATH — the first resource for mathematics

Global behavior of a class of three-dimensional competitive systems. (English) Zbl 0838.34060
The authors consider the three-dimensional competitive system (1) \(\dot X= F(X)\), \(F\in C^2: R^3\to R^3\), where \(X= (x_1, x_2, x_3)\), \(F= (f_1, f_2, f_3)\), \(\partial f_i(x)/\partial x_j\leq 0\), \(i\neq j\). The existence of invariant surfaces and the existence of semi-stable periodic orbits of the system (1) is proved.
MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Levine, D. S.,Qualitative theory of a third-order nonlinear system with examples in population dynamics and chemical kinetics, Mathematical Biosciences,77(1985), 17–33. · Zbl 0579.92020
[2] Yuan, Jiancheng,Qualitative analysis of a class of competitive and cooperative systems, Acta Mathematica Sinica, New Series,4:2 (1988), 124–141. · Zbl 0667.93055
[3] Grassman, W.,Periodic solutions of autonomous differential equations in higher dimensional spaces, Rocky Mountain J. Math.,7(3)(1977), 457–466. · Zbl 0373.34021
[4] Tang, Baorong,On the existence of periodic solutions in the limited explodator model for the Belousov-Zhabotinskii reaction, Nonlinear Analysis, T. M. A.,13(1989), 1359–1374. · Zbl 0686.34039
[5] Hirsch, M. W.,Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal.,13(1982), 167–179. · Zbl 0494.34017
[6] —-,Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal.,16(1985), 423–429. · Zbl 0658.34023
[7] —-,Systems of differential equations that are competitive or cooperative. III: Competing species, Nonlinearity,1(1988), 51–71. · Zbl 0658.34024
[8] –,Systems of differential equations that are competitive or cooperative. IV: Structural stability in 3-dimensional systems, SIAM J. Math. Anal., in press. · Zbl 0734.34042
[9] —-,Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems, J. Diff. Equats.,80(1989), 94–106. · Zbl 0712.34045
[10] Smith, H. L.,Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev.,30(1988), 87–113. · Zbl 0674.34012
[11] —-,Periodic orbits of competitive and cooperative systems, J. Diff. Equats.,65(1986), 361–373. · Zbl 0615.34027
[12] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations, D. C. Heath, Boston, 1965. · Zbl 0154.09301
[13] May, R. M. & Leonard, W.J.,Nonlinear aspects of competition between three species, SIAM J. Appl. Math.,29(1975), 243–253. · Zbl 0314.92008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.