On peaks in carrying simplices

• Autorzy: Janusz Mierczyński
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 81
• Numer: 2
• Strony: 285-292

Daty

wydano

1999

otrzymano

1999-03-08

Twórcy

autor

Janusz Mierczyński

• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

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