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)
Traveling waves in a diffusive predator-prey model with Holling type-III functional response. (English) Zbl 1155.37046
The authors establish the existence of travelling wave solutions and small amplitude travelling wave train solutions for a reaction-diffusion model based on a predator-prey system with a Holling type III functional response. The proof used the shooting argument, invariant manifold theory and Hopf bifurcation theorem. For simplicity It is assumed that the prey is not diffusive but this condition may be relaxed. This reviewer likes to add that Holling type III is useful in modeling the immune system hence the model studied in this paper may be used to model immune system-antigen interaction.

37N25Dynamical systems in biology
92D25Population dynamics (general)
Full Text: DOI
[1] Berryman, A. A.: The origins and evolution of predator -- prey theory, Ecology 73, 1530-1535 (1992)
[2] Chow, P. L.; Tam, W. C.: Periodic and traveling wave solutions to Volterra -- Lotka equations with diffusion, Bull math biol 12, 643-658 (1976) · Zbl 0345.92007
[3] Dunbar, S.: Traveling wave solutions of diffusive Lotka -- voterra equations, J math biol 17, 11-32 (1983) · Zbl 0509.92024 · doi:10.1007/BF00276112
[4] Dunbar, S.: Traveling wave solutions of diffusive Lotka -- voterra equations: A heteroclinic connection in R4, Trans am math soc 286, 557-594 (1984) · Zbl 0556.35078 · doi:10.2307/1999810
[5] Dunbar, S.: Traveling wave solutions of diffusive predator -- prey equations: periodic orbits and point-to periodic hertoclinic orbits, SIAM J appl math 46, 1057-1078 (1986) · Zbl 0617.92020 · doi:10.1137/0146063
[6] Fan, Y. H.; Li, W. T.: Global asymptotic stability of a ratio-dependent predator -- prey system with diffusion, J comput appl math 188, 205-227 (2005) · Zbl 1093.35039 · doi:10.1016/j.cam.2005.04.007
[7] Fan, Y. H.; Li, W. T.; Wang, L. L.: Periodic solutions of delayed ratio-dependent predator -- prey models with monotonic or nonmonotonic functional responses, Nonlinear anal RWA 5, 247-263 (2004) · Zbl 1069.34098 · doi:10.1016/S1468-1218(03)00036-1
[8] Freedman, H. I.: Deterministic mathematical models in population ecology, (1980) · Zbl 0448.92023
[9] Gardner, R.: Existence of traveling wave solutions of predator -- prey systems via the connection index, SIAM J appl math 44, 56-79 (1984) · Zbl 0541.35044 · doi:10.1137/0144006
[10] Gourley, S. A.; Britton, N. F.: A predator -- prey reaction-diffusion system with nonlocal effects, J math biol 34, 297-333 (1996) · Zbl 0840.92018
[11] Hartman, P.: Ordinary differential equations, (1973) · Zbl 0281.34001
[12] Huang, J.; Lu, G.; Ruan, S.: Existence of traveling wave solutions in a diffusive predator -- prey model, J math biol 46, 132-152 (2003) · Zbl 1018.92026 · doi:10.1007/s00285-002-0171-9
[13] Huo, H. F.; Li, W. T.: Periodic solution of a delayed predator -- prey system with michaelis -- menten type functional response, J comput appl math 166, 453-463 (2004) · Zbl 1047.34081 · doi:10.1016/j.cam.2003.08.042
[14] Jiang, G.; Lu, Q.; Qian, L.: Complex dynamics of a Holling type II prey -- predator system with state feedback control, Chaos, solitons & fractals 31, 448-461 (2007) · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077
[15] Lasalle, J. P.: Stability theory for ordinary differential equations, J diff eqns 4, 57-65 (1968) · Zbl 0159.12002 · doi:10.1016/0022-0396(68)90048-X
[16] May, R.: Stablity and complexity in model ecosystems, (1974)
[17] Mischaikow, K.; Reineck, J. F.: Traveling waves in predator -- prey systems, SIAM J math anal 24, 987-1008 (1993) · Zbl 0815.35045 · doi:10.1137/0524068
[18] Murray, J. D.: Mathematical biology, (1998) · Zbl 0704.92001
[19] Owen, M. R.; Lewis, M. A.: How predation can solw stop or reverse a prey invasion, Bull math biol 63, 655-684 (2001) · Zbl 1323.92181
[20] Seydel, R.: Practical bifurcation and stability analysis: from equilibrium to chaos, (1994) · Zbl 0806.34028
[21] Volpert AI, Volpert VA, Volpert VA. ”Travelling wave solutions of parabolic systems” translations of mathematical monographs. Vol. 140 Am Math Soc Providence RI 1994.
[22] Wang, L. L.; Li, W. T.: Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator -- prey model with Holling type functional response, J comput appl math 162, 341-357 (2004) · Zbl 1076.34085 · doi:10.1016/j.cam.2003.06.005
[23] Xu, R.; Chaplain, M. A. J.; Davidson, F. A.: Travelling wave and convergence in stage-structured reaction-diffusion competitive models with nonlocal delays, Chaos, solitons & fractals 30, 974-992 (2006) · Zbl 1142.35477 · doi:10.1016/j.chaos.2005.09.022
[24] Zhang, S.; Tan, D.; Chen, L.: Chaotic behavior of a chemostat model with beddington -- deangelis functional response and periodically impulsive invasion, Chaos, solitons & fractals 29, 474-482 (2006) · Zbl 1121.92070 · doi:10.1016/j.chaos.2005.08.026