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)
Periodicity and blowup in a two-species cooperating model. (English) Zbl 1202.35115
Summary: The cooperating two-species Lotka-Volterra model is discussed. Existence and asymptotic behavior of $T$-periodic solutions for a periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are first investigated. The blowup properties of solutions for the same system are given then. It is shown that periodic solutions exist if the intra-specific competitions are strong whereas blowup solutions exist under certain conditions if the intra-specific competitions are weak. Numerical simulations and a brief discussion are also presented in the last section.

35K51Second-order parabolic systems, initial bondary value problems
92D25Population dynamics (general)
35K57Reaction-diffusion equations
35B10Periodic solutions of PDE
35K58Semilinear parabolic equations
35B44Blow-up (PDE)
Full Text: DOI
[1] Gan, W. Z.; Lin, Z. G.: Coexistence and asymptotic periodicity in a competitor--competitor--mutualist model, J. math. Anal. appl. 337, No. 2, 1089-1099 (2008) · Zbl 1139.35050 · doi:10.1016/j.jmaa.2007.04.022
[2] Tian, C. R.; Lin, Z. G.: Periodic solutions of reaction diffusion systems in a half-space domain, Nonlinear anal. Real world appl. 9, No. 3, 811-821 (2008) · Zbl 1146.35305 · doi:10.1016/j.nonrwa.2007.01.001
[3] Xu, D. S.; Zhao, X. Q.: Dynamics in a periodic competitive model with stage structure, J. math. Anal. appl. 311, No. 2, 417-438 (2005) · Zbl 1077.37051 · doi:10.1016/j.jmaa.2005.02.062
[4] Zhang, Q. Y.; Lin, Z. G.: Periodic solutions of quasilinear parabolic systems with nonlinear boundary conditions, Nonlinear anal. 72, No. 7--8, 3429-3435 (2010) · Zbl 1195.35029 · doi:10.1016/j.na.2009.12.026
[5] Feng, W.; Lu, X.: Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear anal. 26, No. 2, 171-178 (1996) · Zbl 0842.35129 · doi:10.1016/0362-546X(94)00271-I
[6] Hess, P.: Periodic--parabolic boundary value problems and positivity, Pitman research notes in mathematics 247 (1991) · Zbl 0731.35050
[7] Zhao, X. Q.: Global asymptotic behavior in a periodic competitior--competitor--mutualist parabolic system, Nonlinear anal. 29, No. 5, 551-568 (1997) · Zbl 0876.35058 · doi:10.1016/S0362-546X(96)00056-9
[8] Pao, C. V.: Strongly coupled elliptic systems and applications to Lotka--Volterra models with cross-diffusion, Nonlinear anal. 60, No. 7, 1197-1217 (2005) · Zbl 1074.35034 · doi:10.1016/j.na.2004.10.008
[9] Pao, C. V.: Stability and attractivity of periodic solutions of parabolic systems with time delay, J. math. Anal. appl. 304, No. 2, 423-450 (2005) · Zbl 1063.35020 · doi:10.1016/j.jmaa.2004.09.014
[10] Zhou, L.; Fu, Y. P.: Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays, J. math. Anal. appl. 250, No. 1, 139-161 (2000) · Zbl 0970.35004 · doi:10.1006/jmaa.2000.6986
[11] Kim, K. I.; Lin, Z. G.: Blowup in a three-species cooperating model, Appl. math. Lett. 17, No. 1, 89-94 (2004) · Zbl 1047.35055 · doi:10.1016/S0893-9659(04)90017-1
[12] Korman, P.; Leung, A.: On the existence and uniqueness of positive steady states in the Volterra--Lotka ecological models with diffusion, Appl. anal. 44, No. 3--4, 191-207 (1992)
[13] Li, M.; Chen, M. X.: Blowup properties for nonlinear degenerate diffusion equations with nonlocal sources, Nonlinear anal. Real world appl. 11, No. 2, 1122-1130 (2010) · Zbl 1180.35136 · doi:10.1016/j.nonrwa.2009.02.006
[14] Lou, Y.; Nagylaki, T.; Ni, W. M.: On diffusion-induced blowups in a mutualistic model, Nonlinear anal. 45, No. 3, 329-342 (2001) · Zbl 0980.35059 · doi:10.1016/S0362-546X(99)00346-6
[15] Pao, C. V.: Nonlinear parabolic and elliptic equations, (1992) · Zbl 0777.35001
[16] Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N.: Linear and quasilinear equations of parabolic type, (1968) · Zbl 0174.15403
[17] Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations, (1967) · Zbl 0153.13602