# HOMEOMORPHISMS OF THE DISK WITH TRIVIAL DYNAMICS AND EXTINCTION OF COMPETITIVE SYSTEMS

@article{Campos1997HOMEOMORPHISMSOT, title={HOMEOMORPHISMS OF THE DISK WITH TRIVIAL DYNAMICS AND EXTINCTION OF COMPETITIVE SYSTEMS}, author={Juan Campos and Rafael Ortega and Antonio Tineo}, journal={Journal of Differential Equations}, year={1997}, volume={138}, pages={157-170} }

Let h be a homeomorphism of a closed two-dimensional disk D and let Fix(h) denote the set of fixed points of h. We say that h has trivial dynamics if the omega limit set of every orbit [h( p)]n # Z , p # D, is included in Fix(h). In general, the dynamics of a homeomorphism of the disk can be very intricate and one cannot expect this simple behavior unless additional assumptions are imposed. In this paper we shall assume that h is orientation-preserving and that all fixed points are on the…

## 24 Citations

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## References

SHOWING 1-10 OF 11 REFERENCES

### An extension of the results of hirsch on systems of differential equations which are competitive or cooperative

- Mathematics
- 1996

A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The…

### Systems of differential equations which are competitive or cooperative: III. Competing species

- Mathematics
- 1988

For pt.II, see J. Math. Anal., vol.16, p.423 (1985). Persistent trajectories of the n-dimensional system xi=xiNi(x1, . . ., xn), xi>or=0, are studied under the assumptions that the system is…

### Competition systems with periodic coefficients: A geometric approach

- Mathematics
- 1981

The classical two-species competition system is modified to include coefficients which are time-periodic with the same period. We show first that all (nonnegative) solutions converge to a periodic…

### Théoréme de translation plane de brouwer et généralisations du théoréme de Poincaré-Birkhoff

- Mathematics
- 1994

### On the differential equations of species in competition

- BiologyJournal of mathematical biology
- 1976

SummaryIt is shown that the ordinary differential equation commonly used to describe competing species are compatible with any dynamical behavior provided the number of species in very large.

### The ore me de translation plane de Brouwer et ge ne ralisations du the ore me de Poincare Birkhoff, Topology

- 1994

### On the Number of Positive Periodic Solutions for Planar Competing Lotka-Volterra Systems

- Mathematics
- 1995

### By:DS . Date:03:07:01 . Time:04:41 LOP8M. V8.0. Page 01:01 Codes: 2669 Signs: 960 . Length: 45 pic 0 pts, 190 mm REFERENCES

- By:DS . Date:03:07:01 . Time:04:41 LOP8M. V8.0. Page 01:01 Codes: 2669 Signs: 960 . Length: 45 pic 0 pts, 190 mm REFERENCES

### On the number of positive periodic solutions for planar competing Lokta

- Volterra systems, J. Math. Anal. Appl
- 1995