×

zbMATH — the first resource for mathematics

Asymptotic properties and simulations of a stochastic logistic model under regime switching II. (English) Zbl 1255.60129
Summary: This is a continuation of our paper [Math. Comput. Modelling 54, No. 9–10, 2139–2154 (2011; Zbl 1235.60099)]. First, we establish the sufficient conditions for stochastic permanence which are much weaker than those in our previous paper. Then we study some new asymptotic properties of this model. The lower-growth rate and the upper-growth rate of the positive solution are investigated. The superior limit of the average in time of the sample path of the solution is also estimated. Finally, some simulation figures are introduced to illustrate the main results. Some recent investigations are improved and generalized.

MSC:
60J28 Applications of continuous-time Markov processes on discrete state spaces
34D05 Asymptotic properties of solutions to ordinary differential equations
92D40 Ecology
34F05 Ordinary differential equations and systems with randomness
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gilpin, M.E.; Ayala, F.G., Global models of growth and competition, Proc. natl. acad. sci. USA, 70, 3590-3593, (1973) · Zbl 0272.92016
[2] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[3] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002
[4] Faria, T., Asymptotic stability for delayed logistic type equations, Math. comput. modelling, 43, 433-445, (2006) · Zbl 1145.34043
[5] Li, Z.; Chen, F., Almost periodic solutions of a discrete almost periodic logistic equation, Math. comput. modelling, 50, 254-259, (2009) · Zbl 1185.39011
[6] Li, H.; Muroyab, Y.; Nakata, Y.; Yuan, R., Global stability of nonautonomous logistic equations with a piecewise constant delay, Nonlinear anal. real world appl., 11, 2115-2126, (2010) · Zbl 1196.34108
[7] Gard, T.C., Introduction to stochastic differential equations, (1988), Dekker New York · Zbl 0682.92018
[8] Samanta, G.P.; Chakrabarti, C.G., On stability and fluctuation in Gompertzian and logistic growth models, Appl. math. lett., 3, 119-121, (1990) · Zbl 0707.92021
[9] Samanta, G.P., Influence of environmental noise in Gompertzian growth model, J. math. phys. sci., 26, 503-511, (1992) · Zbl 0778.92017
[10] Samanta, G.P., Logistic growth under coloured noise, Bull. math. de la soc. sci. math. de roumanie, 37, 115-122, (1993) · Zbl 0840.92019
[11] Arnold, L.; Horsthemke, W.; Stucki, J.W., The influence of external real and white noise on the lotka – volterra model, Biomed. J., 21, 451, (1979) · Zbl 0433.92019
[12] Beddington, J.R.; May, R.M., Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465, (1977)
[13] Mao, X.R.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in populations dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[14] 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
[15] Luo, Q.; Mao, X.R., Stochastic population dynamics under regime switching, J. math. anal. appl., 334, 69-84, (2007) · Zbl 1113.92052
[16] Luo, Q.; Mao, X.R., Stochastic population dynamics under regime switching II, J. math. anal. appl., 355, 577-593, (2009) · Zbl 1162.92032
[17] Rudnicki, R.; Pichor, K., Influence of stochastic perturbation on prey-predator systems, Math. biosci., 206, 108-119, (2007) · Zbl 1124.92055
[18] Jiang, D.; Shi, N.; Li, X.Y., 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
[19] Li, X.Y.; Mao, X.R., Population dynamical behavior of non-autonomous lotka – volterra competitive system with random perturbation, Discrete contin. dyn. syst., 24, 523-545, (2009) · Zbl 1161.92048
[20] Li, X.Y.; Jiang, D.Q.; Mao, X.R., Population dynamical behavior of lotka – volterra system under regime switching, J. comput. appl. math., 232, 427-448, (2009) · Zbl 1173.60020
[21] Li, X.Y.; Gray, A.; Jiang, D.; Mao, X., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. math. anal. appl., 376, 11-28, (2011) · Zbl 1205.92058
[22] Zhu, C.; Yin, G., On hybrid competitive lotka – volterra ecosystems, Nonlinear anal., 71, e1370-e1379, (2009) · Zbl 1238.34059
[23] Zhu, C.; Yin, G., On competitive lotka – volterra model in random environments, J. math. anal. appl., 357, 154-170, (2009) · Zbl 1182.34078
[24] Krstic, M.; Jovanovic, M., On stochastic population model with the allee effect, Math. comput. modelling, 52, 370-379, (2010) · Zbl 1201.60069
[25] 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)
[26] Liu, M.; Wang, K., Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II, J. theoret. biol., 267, 283-291, (2010)
[27] Liu, M.; Wang, K., Extinction and permanence in a stochastic non-autonomous population system, Appl. math. lett., 23, 1464-1467, (2010) · Zbl 1206.34079
[28] Liu, M.; Wang, K.; Wu, Q., Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. math. biol., 73, 1969-2012, (2011) · Zbl 1225.92059
[29] Liu, M.; Wang, K., Persistence and extinction in stochastic non-autonomous logistic systems, J. math. anal. appl., 375, 443-457, (2011) · Zbl 1214.34045
[30] Liu, M.; Wang, K., Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. comput. modelling, 54, 2139-2154, (2011) · Zbl 1235.60099
[31] Mao, X.; Yin, G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273, (2007) · Zbl 1111.93082
[32] Mao, X.R.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.