zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global qualitative analysis for a predator-prey system with delay. (English) Zbl 1145.34042

This paper concerns the following predator-prey system with delay,

dx(t) dt=x(t)[r 1 -a 11 x(t)-a 12 y(t)],dy(t) dt=y(t)[-r 2 +a 21 x(t)-a 22 y(t-τ)]·(*)

Under the assumption that

r 1 a 21 -r 2 a 11 >0,(H)

system (*) has a unique positive equilibrium E * =(r 1 a 22 +r 2 a 12 a 11 a 22 +a 12 a 21 ,r 1 a 21 -r 2 a 11 a 11 a 22 +a 12 a 21 ). When τ=0, it is known that E * is globally asymptotically stable. Also, Wendi Wang and Zhien Ma showed that solutions of (*) are bounded, uniformly persistent and the delay is harmless for the uniform persistence [J. Math. Anal. Appl. 158, 256–268 (1991; Zbl 0731.34085)].

First, by constructing suitable Lyapunov functionals, the authors provide a set of sufficient conditions on the global stability of E * . Notice that the conditions can not be reduced to those for the case where τ=0. Then, it was shown that local Hopf bifurcation can occur. Moreover, the direction, stability and period of the periodic solution bifurcating from E * was investigates through the normal form theorem and center manifold argument [R. D. Nussbaum, Ann. Mat. Pura Appl. IV. Ser. 101, 263–306 (1974; Zbl 0323.34061)]. Finally, using the global Hopf bifurcation theorem due to J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], the authors also investigates the global existence of a periodic solution of (*). This may be the most important contribution of this paper as results on global existence of periodic solutions are scarce in the literature.

34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)