On peaks in carrying simplices

Janusz Mierczyński

Colloquium Mathematicae (1999)

  • Volume: 81, Issue: 2, page 285-292
  • ISSN: 0010-1354

Abstract

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A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka-Volterra type that result translates into a simple condition on the coefficients.

How to cite

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Mierczyński, Janusz. "On peaks in carrying simplices." Colloquium Mathematicae 81.2 (1999): 285-292. <http://eudml.org/doc/210740>.

@article{Mierczyński1999,
abstract = {A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka-Volterra type that result translates into a simple condition on the coefficients.},
author = {Mierczyński, Janusz},
journal = {Colloquium Mathematicae},
keywords = {invariant manifold; repeller; dissipative; Lotka-Volterra type},
language = {eng},
number = {2},
pages = {285-292},
title = {On peaks in carrying simplices},
url = {http://eudml.org/doc/210740},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Mierczyński, Janusz
TI - On peaks in carrying simplices
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 2
SP - 285
EP - 292
AB - A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka-Volterra type that result translates into a simple condition on the coefficients.
LA - eng
KW - invariant manifold; repeller; dissipative; Lotka-Volterra type
UR - http://eudml.org/doc/210740
ER -

References

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  1. [1] E. Akin, The General Topology of Dynamical Systems, Grad. Stud. Math. 1, Amer. Math. Soc., Providence, RI, 1993. Zbl0781.54025
  2. [2] M. Benaïm, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 137 (1997), 302-319. Zbl0889.58013
  3. [3] P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math. 446 (1994), 115-135. Zbl0783.58061
  4. [4] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, RI, 1978. 
  5. [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Mono- graphs 25, Amer. Math. Soc., Providence, RI, 1988. 
  6. [6] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), 51-71. Zbl0658.34024
  7. [7] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, Berlin, 1977. 
  8. [8] J. Mierczyński, The C 1 property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations 111 (1994), 385-409. Zbl0804.34048
  9. [9] J. Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc. 127 (1999), 543-551. Zbl0912.34037
  10. [10] J. Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: A counterexample, Dynam. Contin. Discrete Impuls. Systems 6 (1999), 149-154. 
  11. [11] --, Smoothness of unordered invariant curves for two-dimensional strongly competitive systems, Appl. Math. (Warsaw) 25 (1999), 449-455. Zbl1005.34042
  12. [12] I. Tereščák, Dynamics of C 1 smooth strongly monotone discrete-time dynamical systems, preprint. 
  13. [13] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems 8 (1993), 189-217. Zbl0797.92025

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