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)
Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. (English) Zbl 1205.92058
Summary: We prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples.
MSC:
92D25Population dynamics (general)
60J20Applications of Markov chains and discrete-time Markov processes on general state spaces
60J70Applications of Brownian motions and diffusion theory
References:
[1]Ahmad, S.; Lazer, A. C.: Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear anal. 40, 37-49 (2000) · Zbl 0955.34041 · doi:10.1016/S0362-546X(00)85003-8
[2]Mao, X.; Marion, G.; Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[3]Mao, X.; Marion, G.; Renshaw, E.: Asymptotic behavior 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
[4]Mao, X.: Delay population dynamics and environmental noise, Stoch. dyn. 5, No. 2, 149-162 (2005) · Zbl 1093.60033 · doi:10.1142/S021949370500133X
[5]Jiang, D.; Shi, N.: A note on nonautonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 164-172 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027
[6]Jiang, D.; Shi, N.; Li, X.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 340, 588-597 (2008) · Zbl 1140.60032 · doi:10.1016/j.jmaa.2007.08.014
[7]Gard, T. C.: Persistence in stochastic food web models, Bull. math. Biol. 46, 357-370 (1984) · Zbl 0533.92028
[8]Gard, T. C.: Stability for multispecies population models in random environments, Nonlinear anal. 10, 1411-1419 (1986) · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[9]Gard, T. C.: Introduction to stochastic differential equations, (1998)
[10]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
[11]Bahar, A.; Mao, X.: Stochastic delay population dynamics, Int. J. Pure appl. Math. 11, 377-400 (2004) · Zbl 1043.92028
[12]Pang, S.; Deng, F.; Mao, X.: Asymptotic property of stochastic population dynamics, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 15, 602-620 (2008) · Zbl 1171.34038
[13]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, 938-957 (2006) · Zbl 1113.34042 · doi:10.1016/j.jmaa.2005.11.009
[14]Luo, Q.; Mao, X.: Stochastic population dynamics under regime switching, J. math. Anal. appl. 334, 69-84 (2007) · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032
[15]Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y.: Dynamical behavior 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 · doi:10.1016/j.cam.2004.02.001
[16]Slatkin, M.: The dynamics of a population in a Markovian environment, Ecology 59, 249-256 (1978)
[17]Freedman, H. I.; Ruan, S.: Uniform persistence in functional differential equations, J. differential equations 115, 173-192 (1995) · Zbl 0814.34064 · doi:10.1006/jdeq.1995.1011
[18]Mao, X.: Stochastic differential equations and applications, (1997)
[19]Mao, X.; Yuan, C.: Stochastic differential equations with Markovian switching, (2006) · Zbl 1109.60043 · doi:10.1155/JAMSA/2006/59032
[20]Mao, X.; Yin, G.; Yuan, C.: Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica 43, 264-273 (2007) · Zbl 1111.93082 · doi:10.1016/j.automatica.2006.09.006