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)
Bifurcations of a predator-prey system of Holling and Leslie types. (English) Zbl 1156.34029

The authors study the following predator-prey model with Holling type-IV functional response and Leslie type numerical response for the predator

x ˙(t)=rx(t)1-x(t) K-mx(t)y(t) b+x 2 (t),y ˙(t)=y(t)s1-y(t) hx(t),(1)

where x(t) and y(t) represent the densities of the prey and the predator population at time t, respectively. r,K,b,s and h are positive parameters. It is shown that system (1) admits two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of codimension 2 and the other is a multiple focus of multiplicity one. When the parameters vary in a small neighborhood of the values of parameters, system (1) undergoes a Bogdanov-Takens bifurcation and a subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further derived that by choosing different values of parameters, system (1) can have a stable limit cycle enclosing two equilibria, or an unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. It is also proved that system (1) never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters.

MSC:
34C23Bifurcation (ODE)
92D25Population dynamics (general)
37G15Bifurcations of limit cycles and periodic orbits
34C05Location of integral curves, singular points, limit cycles (ODE)