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)
Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion. (English) Zbl 1238.35162
Summary: A Lotka-Volterra type predator-prey elliptic system with nonlinear cross-diffusion is considered, representing the tendency of predators to get closer to prey under Robin boundary condition. The necessary and sufficient conditions for the existence of coexistence states are given in terms of principal eigenvalues, involving semi-trivial solutions using the degree theory. This paper also discusses the ecological meaning and effect of decreasing cross-diffusion induced on predators by prey in a specific model.

MSC:
35Q92PDEs in connection with biology and other natural sciences
92D25Population dynamics (general)
WorldCat.org
Full Text: DOI
References:
[1] Blat, J.; Brown, K. J.: Bifurcation of steady-state solutions in predator--prey and competition systems. Proc. roy. Soc. Edinburgh sect. A 97, 21-34 (1984) · Zbl 0554.92012
[2] Dancer, E. N.: On uniqueness and stability for solutions of singularly perturbed predator--prey type equations with diffusion. J. differential equations 102, No. 1, 1-32 (1993) · Zbl 0817.35042
[3] Li, L.: Coexistence theorems of steady states for predator--prey interacting systems. Trans. amer. Math. soc. 305, 143-166 (1988) · Zbl 0655.35021
[4] Li, L.: On positive solutions of a nonlinear equilibrium boundary value problem. J. math. Anal. appl. 138, 537-549 (1989) · Zbl 0682.35040
[5] Li, L.; Logan, R.: Positive solutions to general elliptic competition models. Differential integral equations 4, 817-834 (1991) · Zbl 0751.35014
[6] López-Gómez, J.; Pardo, R.: Existence and uniqueness of coexistence states for the predator--prey model with diffusion: the scalar case. Differential integral equations 6, No. 5, 1025-1031 (1993) · Zbl 0813.34022
[7] Shigesada, N.; Kawasaki, K.; Teramoto, E.: Spatial segregation of interacting species. J. theoret. Biol. 79, 83-99 (1979)
[8] Aronson, D. G.; Tesei, A.; Weinberger, H.: A density-dependent diffusion system with stable discontinuous stationary solutions. Ann. mat. Pura appl. (4) 152, 259-280 (1988) · Zbl 0673.35054
[9] Lou, Y.; Ni, W. M.: Diffusion, self-diffusion and cross-diffusion. J. differential equations 131, 79-131 (1996) · Zbl 0867.35032
[10] Lou, Y.; Ni, W. M.: Diffusion vs cross-diffusion: an elliptic approach. J. differential equations 154, No. 1, 157-190 (1999) · Zbl 0934.35040
[11] Ruan, W. H.: Positive steady-state solutions of a competing reaction--diffusion system with large cross-diffusion coefficients. J. math. Anal. appl. 197, 558-578 (1996) · Zbl 0855.35066
[12] Ryu, K.; Ahn, I.: Positive steady-states for two interacting species models with linear self-cross diffusions. Discrete contin. Dyn. syst. 9, No. 4, 1049-1061 (2003) · Zbl 1065.35119
[13] Ryu, K.; Ahn, I.: Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics. J. math. Anal. appl. 283, No. 1, 46-65 (2003) · Zbl 1115.35321
[14] Okubo, A.; Levin, S. A.: Diffusion and ecological problems: modern perspectives. (2001) · Zbl 1027.92022
[15] Chen, X. F.; Qi, Y. W.; Wang, M. X.: A strongly coupled predator--prey system with non-monotonic functional response. Nonlinear anal. 67, No. 6, 1966-1979 (2007) · Zbl 05169025
[16] Pang, P. Y. H.; Wang, M. X.: Strategy and stationary pattern in a three-species predator--prey model. J. differential equations 200, No. 2, 245-273 (2004) · Zbl 1106.35016
[17] Wang, M. X.: Stationary patterns of strongly coupled prey--predator models. J. math. Anal. appl. 292, No. 2, 484-505 (2004) · Zbl 1160.35325
[18] Wang, M. X.: Stationary patterns caused by cross-diffusion for a three-species prey--predator model. Comput. math. Appl. 52, No. 5, 707-720 (2006) · Zbl 1121.92069
[19] Kadota, T.; Kuto, K.: Positive steady states for a prey--predator model with some nonlinear diffusion terms. J. math. Anal. appl. 323, No. 2, 1387-1401 (2006) · Zbl 1160.35441
[20] Kuto, K.: A strongly coupled diffusion effect on the stationary solution set of a prey--predator model. Adv. differential equations 12, No. 2, 145-172 (2007) · Zbl 1167.35048
[21] K. Kuto, Y. Yamada, Limiting characterization of stationary solutions for a prey--predator model with nonlinear diffusion of fractional type, preprint · Zbl 1240.35157
[22] Cano-Casanova, S.: Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems. Nonlinear anal. Ser. A: theory methods 49, No. 3, 361-430 (2002) · Zbl 1142.35509
[23] Cano-Casanova, S.; López-Gómez, J.: Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. J. differential equations 178, No. 1, 123-211 (2002) · Zbl 1086.35073
[24] Amann, H.; López-Gómez, J.: A priori bounds and multiple solutions for superlinear indefinite elliptic problems. J. differential equations 146, No. 2, 336-374 (1998) · Zbl 0909.35044
[25] Pao, C. V.: On nonlinear parabolic and elliptic equations. (1992) · Zbl 0777.35001
[26] Dancer, E. N.: On the indices of fixed points of mappings in cones and applications. J. math. Anal. appl. 91, 131-151 (1983) · Zbl 0512.47045
[27] López-Gómez, J.: Positive periodic solutions of Lotka--Volterra reaction--diffusion systems. Differential integral equations 5, No. 1, 55-72 (1992) · Zbl 0754.35065
[28] Wang, M.; Li, Z. Y.; Ye, Q. X.: Existence of positive solutions for semilinear elliptic system. School on qualitative aspects and applications of nonlinear evolution equations (Trieste1990), 256-259 (1991)
[29] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044
[30] Nakashima, K.; Yamada, Y.: Positive steady states for prey--predator models with cross-diffusion. Adv. differential equations 1, No. 6, 1099-1122 (1996) · Zbl 0863.35034