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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
\[ dx(t) = x(t)[r(t) -a(t)x(t)]dt+ \sigma(t)x^{2}(t)dB(t) \]
and
\[ dx(t) = x(t)[r(t) -a(t)x(t)]dt+ \sigma(t)x(t)dB(t). \]
Carrying out the survival analysis, they find sufficient conditions for extinction and examine the questions of persistence and permanence.

MSC:
34F05 Ordinary differential equations and systems with randomness
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
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[1] May, R.M., Stability and complexity in model ecosystems, (1973), Princeton Univ. Press
[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
[3] Golpalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht
[4] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · 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
[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
[8] Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 913-936, (2005) · Zbl 1078.34035
[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
[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
[11] Bahar, A.; Mao, X., Stochastic delay Lotka-Volterra model, J. math. anal. appl., 292, 364-380, (2004) · Zbl 1043.92034
[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
[13] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J. math. anal. appl., 334, 69-84, (2007) · Zbl 1113.92052
[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
[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
[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
[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
[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
[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
[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) · Zbl 1406.92673
[24] Hallam, T.G.; Ma, Z., Persistence in population models with demographic fluctuations, J. math. biol., 24, 327-339, (1986) · Zbl 0606.92022
[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
[26] Ma, Z.; Hallam, T.G., Effects of parameter fluctuations on community survival, Math. biosci., 86, 35-49, (1987) · Zbl 0631.92019
[27] Wang, W.; Ma, Z., Permanence of a nonautomonous population model, Acta math. appl. sin. engl. ser., 1, 86-95, (1998) · Zbl 0940.92020
[28] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publishing Chichester · Zbl 0874.60050
[29] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007
[30] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002
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