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Hopf bifurcations in a three-species food chain system with multiple delays

100%

Xie X. , Zhang W.

Open Mathematics

2017

tom 15

nr 1

508-519

This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

On super-weakly compact sets and uniformly convexifiable sets

80%

Cheng L. , Cheng Q. , Wang B. , Zhang W.

Studia Mathematica

2010

tom 199

nr 2

145-169

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis-Figiel-Johnson-Pełczyński lemma.