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Asymptotic properties and simulations of a stochastic logistic model under regime switching. (English) Zbl 1235.60099
Summary: Taking both white noise and colored environmental noise into account, a general stochastic logistic model under regime switching is proposed and studied. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, stochastic permanence and global attractivity are established. The critical number between weak persistence and extinction is obtained. Moreover, some simulation figures are introduced to illustrate the main results.

MSC:
60J28Applications of continuous-time Markov processes on discrete state spaces
92D40Ecology
34D05Asymptotic stability of ODE
34F05ODE with randomness
References:
[1]May, R. M.: Stability and complexity in model ecosystems, (2001)
[2]Golpalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[3]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[4]Faria, T.: Asymptotic stability for delayed logistic type equations, Math. comput. Modelling 43, 433-445 (2006) · Zbl 1145.34043 · doi:10.1016/j.mcm.2005.11.006
[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 · doi:10.1016/j.mcm.2008.12.017
[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 · doi:10.1016/j.nonrwa.2009.06.003
[7]Gard, T. C.: Introduction to stochastic differential equations, (1988)
[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 · doi:10.1016/0893-9659(90)90153-3
[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]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
[12]Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y.: Dynamical behaviour of Lotka–Volterra competition systems: nonautonomous 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
[13]Jeffries, C.: Stability of predation ecosystem models, Ecology 57, 1321-1325 (1976)
[14]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
[15]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 · doi:10.1002/bimj.4710210507
[16]Beddington, J. R.; May, R. M.: Harvesting natural populations in a randomly fluctuating environment, Science 197, 463-465 (1977)
[17]Rudnicki, R.; Pichor, K.: Influence of stochastic perturbation on prey-predator systems, Math. biosci. 206, 108-119 (2007) · Zbl 1124.92055 · doi:10.1016/j.mbs.2006.03.006
[18]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
[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 · doi:10.3934/dcds.2009.24.523
[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 · doi:10.1016/j.cam.2009.06.021
[21]Luo, Q.; Mao, X. R.: Stochastic population dynamics under regime switching II, J. math. Anal. appl. 355, 577-593 (2009) · Zbl 1162.92032 · doi:10.1016/j.jmaa.2009.02.010
[22]Zhu, C.; Yin, G.: On hybrid competitive Lotka–Volterra ecosystems, Nonlinear anal. 71 (2009)
[23]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
[24]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)
[25]Krstic, M.; Jovanovic, M.: On stochastic population model with the allee effect, Math. comput. Modelling 52, 370-379 (2010) · Zbl 1201.60069 · doi:10.1016/j.mcm.2010.02.051
[26]Liu, M.; Wang, K.; Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. math. Biol. (2010)
[27]Liu, M.; Wang, K.: Extinction and permanence in a stochastic nonautonomous population system, Appl. math. Lett. 23, 1464-1467 (2010) · Zbl 1206.34079 · doi:10.1016/j.aml.2010.08.012
[28]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)
[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 · doi:10.1016/j.jmaa.2010.09.058
[30]Luo, Q.; Mao, X. R.: 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
[31]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 · doi:10.1016/S0304-4149(01)00126-0
[32]Mao, X. R.; 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
[33]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
[34]Gilpin, M. E.; Ayala, F. G.: Global models of growth and competition, Proc. nat. Acad. sci. USA 70, 3590-3593 (1973) · Zbl 0272.92016 · doi:10.1073/pnas.70.12.3590
[35]Liu, H.; Ma, Z.: The threshold of survival for system of two species in a polluted environment, J. math. Biol. 30, 49-51 (1991) · Zbl 0745.92028 · doi:10.1007/BF00168006
[36]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
[37]Mao, X. R.; Yuan, C.: Stochastic differential equations with Markovian switching, (2006) · Zbl 1109.60043 · doi:10.1155/JAMSA/2006/59032
[38]Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991)
[39]Barbalat, I.: Systems dequations differentielles d’osci d’oscillations nonlineaires, Revue roumaine de mathematiques pures et appliquees 4, 267-270 (1959) · Zbl 0090.06601
[40]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