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)
Multiple coexistence states for a prey-predator system with cross-diffusion. (English) Zbl 1205.35116

This paper concerns the nonnegative steady-states of the following parabolic system

u t =Δ[(d 1 +ρ 12 v)u]+u(a 1 -b 1 u-c 1 v)inΩ×(0,+),v t =Δ[(d 2 +ρ 21 u)v]+v(a 2 +b 2 u-c 2 v)inΩ×(0,+),u=v=0,onΩ×(0,+),u(x,0)=u 0 (x)0,v(x,0)=v 0 (x)0,xΩ,

where Ω N (N1) is a bounded domain, ρ 12 , ρ 21 0, a 1 ,α i ,b i ,c i (i=1,2) are positive constants and a 2 .

This is a Lotka-Volterra prey-predator model with cross-diffusion effects. It is shown tha under certain assumptions (on the parameters) the system admits a branch of positive steady-states, which is S or I shaped with respect to a bifurcation parameter. The analysis is based on the bifurcation theory and the Lyapunov-Schmidt procedure.

35J65Nonlinear boundary value problems for linear elliptic equations
35B32Bifurcation (PDE)
92D25Population dynamics (general)