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)
Persistence and extinction in stochastic non-autonomous logistic systems. (English) Zbl 1214.34045

Beginning with a general discussion of the classical non-autonomous logistic equation

dx(t)/dt=x(t)[r(t)-a(t)x(t)],x(0)=x 0 >0,

including definitions of persistence in both the deterministic and the stochastic sense, the authors examine the equations

dx(t)=x(t)[r(t)-a(t)x(t)]dt+σ(t)x 2 (t)dB(t)

and

dx(t)=x(t)[r(t)-a(t)x(t)]dt+σ(t)x(t)dB(t)·

Carrying out the survival analysis, they find sufficient conditions for extinction and examine the questions of persistence and permanence.

MSC:
34F05ODE with randomness
92D25Population dynamics (general)
34D05Asymptotic stability of ODE
References:
[1]May, R. M.: Stability and complexity in model ecosystems, (1973)
[2]Freedman, H. I.; Wu, J.: Periodic solutions of single-species models with periodic delay, SIAM J. Math. anal. 23, 689-701 (1992) · Zbl 0764.92016 · doi:10.1137/0523035
[3]Golpalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[4]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[5]Lisena, B.: Global attractivity in nonautonomous logistic equations with delay, Nonlinear anal. Real world appl. 9, 53-63 (2008) · Zbl 1139.34052 · doi:10.1016/j.nonrwa.2006.09.002
[6]Gard, T. C.: Persistence in stochastic food web models, Bull. math. Biol. 46, 357-370 (1984) · Zbl 0533.92028
[7]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
[8]Bandyopadhyay, M.; Chattopadhyay, J.: Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity 18, 913-936 (2005) · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022
[9]Mao, X.; Marion, G.; Renshaw, E.: Environmental Brownian noise suppresses explosions in populations dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[10]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
[11]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
[12]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
[13]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
[14]Beddington, J. R.; May, R. M.: Harvesting natural populations in a randomly fluctuating environment, Science 197, 463-465 (1977)
[15]Braumann, C. A.: Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. biosci. 177-178, 229-245 (2002) · Zbl 1003.92027 · doi:10.1016/S0025-5564(01)00110-9
[16]Jiang, D. Q.; Shi, N. Z.: A note on non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 164-172 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027
[17]Jiang, D. Q.; Shi, N. Z.; Li, X. Y.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 340, 588-597 (2006) · Zbl 1140.60032 · doi:10.1016/j.jmaa.2007.08.014
[18]Pang, S.; Deng, F.; Mao, X.: Asymptotic properties of stochastic population dynamics, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 15, 603-620 (2008) · Zbl 1171.34038
[19]Zhu, C.; Yin, G.: On competitive Lotka-Volterra model in random environments, J. math. Anal. appl. 357, 154-170 (2009) · Zbl 1182.34078 · doi:10.1016/j.jmaa.2009.03.066
[20]Ji, C. Y.; Jiang, D. Q.; Shi, N. Z.: Analysis of a predator-prey model with modified Leslie-gower and Holling-type II schemes with stochastic perturbation, J. math. Anal. appl. 359, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[21]Li, X.; Mao, X.: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete contin. Dyn. syst. 24, 523-545 (2009) · Zbl 1161.92048 · doi:10.3934/dcds.2009.24.523
[22]Liu, M.; Wang, K.: Survival analysis of stochastic single-species population models in polluted environments, Ecol. model. 220, 1347-1357 (2009)
[23]Liu, M.; Wang, K.: Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. theoret. Biol. 264, 934-944 (2010)
[24]Hallam, T. G.; Ma, Z.: Persistence in population models with demographic fluctuations, J. math. Biol. 24, 327-339 (1986) · Zbl 0606.92022 · doi:10.1007/BF00275641
[25]Ma, Z.; Cui, G.; Wang, W.: Persistence and extinction of a population in a polluted environment, Math. biosci. 101, 75-97 (1990) · Zbl 0714.92027 · doi:10.1016/0025-5564(90)90103-6
[26]Ma, Z.; Hallam, T. G.: Effects of parameter fluctuations on community survival, Math. biosci. 86, 35-49 (1987) · Zbl 0631.92019 · doi:10.1016/0025-5564(87)90062-9
[27]Wang, W.; Ma, Z.: Permanence of a nonautomonous population model, Acta math. Appl. sin. Engl. ser. 1, 86-95 (1998) · Zbl 0940.92020 · doi:10.1007/BF02677353
[28]Mao, X.: Stochastic differential equations and applications, (1997)
[29]Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev. 43, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302
[30]Mao, X.; Yuan, C.: Stochastic differential equations with Markovian switching, (2006) · Zbl 1109.60043 · doi:10.1155/JAMSA/2006/59032