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)
Stability of steady-state solutions to a prey--predator system with cross-diffusion. (English) Zbl 1210.35122
The author investigates a Lotka-Volterra prey-predator model with cross-diffusion effects. The stability of the coexistence steady-states is studied. Criteria on the stability of the coexistence steady-states are obtained. The author proves that the Hopf bifurcation phenomenon on steady-states occurs under certain conditions.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35B35Stability of solutions of PDE
92D25Population dynamics (general)
WorldCat.org
Full Text: DOI
References:
[1] Amann, H.: Dynamic theory of quasilinear parabolic equations--II. Reaction--diffusion systems. Differential integral equations 3, 13-75 (1990) · Zbl 0729.35062
[2] Amann, H.: Hopf bifurcation in quasilinear reaction--diffusion systems. Lecture notes in mathematics 1475, 53-63 (1991) · Zbl 0780.35051
[3] Choi, Y. S.; Lui, R.; Yamada, Y.: Existence of global solutions for the shigesada--kawasaki--teramoto model with weak cross-diffusion. Discrete contin. Dynamic systems 9, 1993-2000 (2003) · Zbl 1029.35116
[4] Choi, Y. S.; Lui, R.; Yamada, Y.: Existence of global solutions for the shigesada--kawasaki--teramoto model with strongly coupled cross-diffusion. Discrete contin. Dynamic systems 10, 719-730 (2004) · Zbl 1047.35054
[5] Crandall, M. G.; Rabinowitz, P. H.: The Hopf bifurcation theorem in infinite dimensions. Arch. rational mech. Anal. 67, 53-72 (1977) · Zbl 0385.34020
[6] Drangeid, A. -K.: The principle of linearized stability for quasilinear parabolic evolution equations. Nonlinear anal. TMA 13, 1091-1113 (1989) · Zbl 0694.35009
[7] Du, Y.; Lou, Y.: S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator--prey model. J. differential equations 144, 390-440 (1998) · Zbl 0970.35030
[8] Grisvard, P.: Charactérisation de quelques espaces d’interpolation. Arch. rational mech. Anal. 25, 40-63 (1967) · Zbl 0187.05901
[9] Kan-On, Y.: Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics. Hiroshima math. J. 23, 509-536 (1993) · Zbl 0792.35086
[10] Kato, T.: Perturbation theory for linear operators. (1966) · Zbl 0148.12601
[11] K. Kuto, Y. Yamada, Multiple coexistence states for a prey--predator system with cross-diffusion, J. Differential Equations, in press. · Zbl 1205.35116
[12] Li, L.: Coexistence theorems of steady states for predator--prey interacting system. Trans. amer. Math. soc. 305, 143-166 (1988) · Zbl 0655.35021
[13] Lions, J. L.; Peetre, J.: Sur une classe d’espaces d’interpolation. Publ. math. IHES 19, 5-68 (1964)
[14] López-Gómez, J.; Pardo, R.: Coexistence regions in Lotka--Volterra models with diffusion. Nonlinear anal. TMA 19, 11-28 (1992) · Zbl 0781.35014
[15] López-Gómez, J.; Pardo, R.: Existence and uniqueness of coexistence states for the predator--prey model with diffusion. Differential integral equations 6, 1025-1031 (1993) · Zbl 0813.34022
[16] Lou, Y.; Ni, W. -M.; Wu, Y.: On the global existence of a cross-diffusion system. Discrete contin. Dynamic systems 4, 193-203 (1998) · Zbl 0960.35049
[17] Mimura, M.; Nishiura, Y.; Tesei, A.; Tsujikawa, T.: Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima math. J. 14, 425-449 (1984) · Zbl 0562.92011
[18] Nakashima, K.; Yamada, Y.: Positive steady states for prey--predator models with cross-diffusion. Adv. differential equations 6, 1099-1122 (1996) · Zbl 0863.35034
[19] A. Okubo, L.A. Levin, Diffusion and Ecological Problems: Modern Perspective, 2nd Edn., Interdisciplinary Applied Mathematics, Vol. 14, Springer, New York, 2001. · Zbl 1027.92022
[20] Potier-Ferry, M.: The linearization principle for the stability of solutions of quasilinear parabolic equations--I. Arch. rational mech. Anal. 77, 301-320 (1981) · Zbl 0497.35006
[21] Shigesada, N.; Kawasaki, K.; Teramoto, E.: Spatial segregation of interacting species. J. theor. Biol. 79, 83-99 (1979)