Download PDF - On peaks in carrying simplices
ArticleOriginal scientific text
Title
On peaks in carrying simplices
Authors 1
Affiliations
- Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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.
Bibliography
- E. Akin, The General Topology of Dynamical Systems, Grad. Stud. Math. 1, Amer. Math. Soc., Providence, RI, 1993.
- M. Benaïm, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 137 (1997), 302-319.
- P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math. 446 (1994), 115-135.
- C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, RI, 1978.
- J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Mono- graphs 25, Amer. Math. Soc., Providence, RI, 1988.
- M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), 51-71.
- M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, Berlin, 1977.
- J. Mierczyński, The
property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations 111 (1994), 385-409. - J. Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc. 127 (1999), 543-551.
- J. Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: A counterexample, Dynam. Contin. Discrete Impuls. Systems 6 (1999), 149-154.
- --, Smoothness of unordered invariant curves for two-dimensional strongly competitive systems, Appl. Math. (Warsaw) 25 (1999), 449-455.
- I. Tereščák, Dynamics of
smooth strongly monotone discrete-time dynamical systems, preprint. - M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems 8 (1993), 189-217.