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)
Stochastic Gilpin-Ayala competition model with infinite delay. (English) Zbl 1206.92070
Summary: We study the stochastic M. E. Gilpin and F. J. Ayala [Proc. Natl. Acad. Sci. USA 70, 3590–3593 (1973; Zbl 0272.92016)] competition model with an infinite delay. We verify that the environmental noise included in the model does not only provide a positive global solution (there is no explosion in a finite time), but this solution is also stochastically ultimately bounded. We obtain certain asymptotic results regarding a large time behavior.
34F05ODE with randomness
34K60Qualitative investigation and simulation of models
[1]Lotka, A.: Elements of physical biology, (1924)
[2]Volterra, V.: Lecons sur la theorie mathematique de la lutte pour la vie, (1931) · Zbl 0002.04202
[3]He, X.; Gopalsamy, K.: Persistence, attractivity, and delay in facultative mutualism, J. math. Anal. appl. 215, 154-173 (1997) · Zbl 0893.34036 · doi:10.1006/jmaa.1997.5632
[4]Bereketoglu, H.; Gyori, I.: Global asymptotic stability in a nonautonomous Lotka – Volterra type system with infinite delay, J. math. Anal. appl. 210, 279-291 (1997) · Zbl 0880.34072 · doi:10.1006/jmaa.1997.5403
[5]Gilpin, M. E.; Ayala, F. J.: Global models of growth and competition, Proc. natl. Acad. sci. USA 70, 3590-3593 (1973) · Zbl 0272.92016 · doi:10.1073/pnas.70.12.3590
[6]Fan, M.; Wang, K.: Global periodic solutions of a generalized n-species gilpin – ayala competition model, Comput. math. Appl. 40, 1141-1151 (2000) · Zbl 0954.92027 · doi:10.1016/S0898-1221(00)00228-5
[7]Mao, X.; Marion, G.; Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics, Stoch. process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[8]Mao, X.; Sabanis, S.; Renshaw, E.: Asymptotic behaviour of the stochastic Lotka – Volterra model, J. math. Anal. appl. 287, 141-156 (2003) · Zbl 1048.92027 · doi:10.1016/S0022-247X(03)00539-0
[9]Du, N. H.; Sam, V. H.: Dynamics of a stochastic Lotka – Volterra model perturbed by white noise, J. math. Anal. appl. 324, 82-97 (2006) · Zbl 1107.92038 · doi:10.1016/j.jmaa.2005.11.064
[10]Lian, B.; Hu, S.: Asymptotic behaviour of the stochastic gilpin – ayala competition models, J. math. Anal. appl. 339, 419-428 (2008) · Zbl 1195.34083 · doi:10.1016/j.jmaa.2007.06.058
[11]Vasilova, M.; Jovanović, M.: Dynamics of gilpin – ayala competition model with random perturbation, Filomat 24, No. 1, 101-113 (2010)
[12]Bahar, A.; Mao, X.: Stochastic delay Lotka – Volterra model, J. math. Anal. appl. 292, 364-380 (2004) · Zbl 1043.92034 · doi:10.1016/j.jmaa.2003.12.004
[13]Wan, L.; Zhou, Q.: Stochastic Lotka – Volterra model with infinite delay, Statist. probab. Lett. 79, 698-706 (2009) · Zbl 1159.92321 · doi:10.1016/j.spl.2008.10.016
[14]Mao, X.; Yuan, C.; Zou, J.: Stochastic differential delay equations of population dynamics, J. math. Anal. appl. 304, 296-320 (2005) · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[15]Lian, B.; Hu, S.: Stochastic delay gilpin – ayala competition models, Stoch. dyn. 6, No. 4, 561-576 (2006) · Zbl 1117.34079 · doi:10.1142/S0219493706001888
[16]Yan, J.: Global positive periodic solutions of periodic n-species competition systems, J. math. Anal. appl. 356, 288-294 (2009) · Zbl 1177.34056 · doi:10.1016/j.jmaa.2009.03.013
[17]Mao, X.: Stochastic differential equations and applications, (2008)
[18]Mao, X.: Exponential stability of stochastic differential equations, (1994)
[19]Sinclair, A. R. E.: Mammal population regulation, keystone processes and ecosystem dynamics, Phil. trans. R. soc. Lond. B 358, 1729-1740 (2003)
[20]Kloeden, P. E.; Platen, E.: Numerical solution of stochastic differential equations, (1995) · Zbl 0858.65148
[21]Song, Y.; Baker, C. T. H.: Qualitative behaviour of numerical approximations to Volterra integro-differential equations, J. comput. Appl. math. 172, 101-115 (2004) · Zbl 1059.65129 · doi:10.1016/j.cam.2003.12.049