zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits. (English) Zbl 0617.92020

The reaction-diffusion system

U t =αU(γ-U)-UW/(1+U),W t =W xx -W+βUW/(1+U)

is considered under some natural assumptions on the positive parameters occuring. This system has drawn much attention even without the diffusive term [see also e.g. H. L. Smith, SIAM J. Appl. Math. 42, 27-43 (1982; Zbl 0489.92019), and the reviewer, Nonlinear Anal., Theory Methods Appl. 8, 1295-1309 (1984; Zbl 0561.92015)].

Assuming the existence of a traveling wave solution the author gets a three-dimensional autonomous system of ordinary differential equations to be satisfied by this solution. Establishing the Hopf bifurcation in the latter system he shows the occurrence of a traveling wave train solution of the original system.

He also shows the existence of a heteroclinic orbit joining an unstable equilibrium (absence of predators) to a positive equilibrium (in the pre- bifurcation situation) and to a periodic orbit (in the post-bifurcation situation).

Reviewer: M.Farkas

MSC:
92D25Population dynamics (general)
35K57Reaction-diffusion equations
37G99Local and nonlocal bifurcation theory
34C05Location of integral curves, singular points, limit cycles (ODE)