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 population dynamics under regime switching. (English) Zbl 1113.92052
Summary: We develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded.

92D25Population dynamics (general)
60H30Applications of stochastic analysis
60J20Applications of Markov chains and discrete-time Markov processes on general state spaces
60H10Stochastic ordinary differential equations
Full Text: DOI
[1] Anderson, W. J.: Continuous-time Markov chains. (1991) · Zbl 0731.60067
[2] Boucher, D. H.: The biology of mutualism. (1985)
[3] Butler, G.; Freedman, H. I.; Waltman, P.: Uniformly persistence systems. Proc. amer. Math. soc. 96, 425-430 (1986) · Zbl 0603.34043
[4] Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y.: Dynamical behaviour of Lotka -- Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise. J. comput. Appl. math. 170, 399-422 (2004) · Zbl 1089.34047
[5] Freedman, H. I.; Ruan, S.: Uniform persistence in functional differential equations. J. differential equations 115, 173-192 (1995) · Zbl 0814.34064
[6] Gilpin, M. E.: Predator -- prey communities. (1975)
[7] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[8] He, X.; Gopalsamy, K.: Persistence, attractivity, and delay in facultative mutualism. J. math. Anal. appl. 215, 154-173 (1997) · Zbl 0893.34036
[9] Hutson, V.; Schmitt, K.: Permanence and the dynamics of biological systems. Math. biosci. 111, 1-71 (1992) · Zbl 0783.92002
[10] Jansen, W.: A permanence theorem for replicator and Lotka -- Volterra system. J. math. Biol. 25, 411-422 (1987) · Zbl 0647.92021
[11] Kirlinger, G.: Permanence of some ecological systems with several predator and one prey species. J. math. Biol. 26, 217-232 (1988) · Zbl 0713.92025
[12] Mao, X.: Stability of stochastic differential equations with respect to semimartingales. (1991) · Zbl 0724.60059
[13] Mao, X.: Exponential stability of stochastic differential equations. (1994) · Zbl 0806.60044
[14] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057
[15] Mao, X.: Stability of stochastic differential equations with Markovian switching. Stochastic process. Appl. 79, 45-67 (1999) · Zbl 0962.60043
[16] Mao, X.; Marion, G.; Renshaw, E.: Environmental noise suppresses explosion in population dynamics. Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046
[17] Skorohod, A. V.: Asymptotic methods in the theory of stochastic differential equations. (1989)
[18] Slatkin, M.: The dynamics of a population in a Markovian environment. Ecology 59, 249-256 (1978)
[19] Takeuchi, Y.: Global dynamical properties of Lotka -- Volterra systems. (1996) · Zbl 0844.34006
[20] Takeuchi, Y.; Du, N. H.; Hieu, N. T.; Sato, K.: Evolution of predator -- prey systems described by a Lotka -- Volterra equation under random environment. J. math. Anal. appl. 323, No. 2, 938-957 (2006) · Zbl 1113.34042
[21] Wolin, C. L.; Lawlor, L. R.: Models of facultative mutualism: density effects. Amer. natural. 124, 843-862 (1984)