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Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting. (English) Zbl 1303.92109
Summary: In this paper, the dynamic behavior of a stage-structured population model involving gestation delay is investigated within stochastically fluctuating environment and harvesting. Firstly, the stability and Hopf bifurcation condition are described on the delayed population model within deterministic environment. Secondly, the stochastic population model systems are discussed by incorporating white noise terms to the deterministic system model. Finally, numerical simulations show that the gestation delay with larger magnitude has ability to drive the system from stable to unstable within the same fluctuating environment and the frequency and amplitude of oscillation for the population density is enhanced as environmental driving forces increase. These indicate that the magnitude of gestation delay plays a crucial role to determine the stability or instability and the magnitude of environmental driving forces plays a crucial role to determine the magnitude of oscillation of the population model system within fluctuating environment.

92D25 Population dynamics (general)
37N25 Dynamical systems in biology
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
Full Text: DOI
[1] Song, X.Y., Chen, L.S.: Modelling and analysis of a single-species system with stage structure and harvesting. Math. Comput. Model. 36, 67–82 (2002) · Zbl 1024.92015 · doi:10.1016/S0895-7177(02)00104-8
[2] Gao, S.J., Chen, L.S., Sun, L.H.: Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos Solitons Fractals 25, 1209–1219 (2005) · Zbl 1065.92056 · doi:10.1016/j.chaos.2004.11.093
[3] Armstrong, C.W., Skonhoft, A.: Marine reserves: a bio-economic model with asymmetric density dependent migration. Ecol. Econ. 57, 466–476 (2006) · doi:10.1016/j.ecolecon.2005.05.010
[4] Sanchirico, J.N., Wilen, J.E.: Optimal spatial management of renewable resources: matching policy scope to ecosystem scale. J. Environ. Econ. Manag. 50, 23–46 (2005) · Zbl 1122.91358 · doi:10.1016/j.jeem.2004.11.001
[5] Jerry, M., Raissi, N.: The optimal strategy for a bioeconomical model of a harvesting renewable resource problem. Math. Comput. Model. 36, 1293–1306 (2002) · Zbl 1077.91036 · doi:10.1016/S0895-7177(02)00277-7
[6] Alvarez, L.H.R., Shepp, L.A.: Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177 (1998) · Zbl 0940.92029 · doi:10.1007/s002850050124
[7] Kar, T.K.: Influence of environmental noises on the Gompertz model of two species fishery. Ecol. Model. 173, 283–293 (2004) · doi:10.1016/j.ecolmodel.2003.08.021
[8] Zhang, X.A., Chen, L.S., Neumann, A.U.: The stage-structured predator-prey model and optimal harvesting policy. Math. Biosci. 168, 201–210 (2000) · Zbl 0961.92037 · doi:10.1016/S0025-5564(00)00033-X
[9] Bandyopadhyay, M., Banerjee, S.: A stage-structured prey-predator model with discrete time delay. Appl. Math. Comput. 182, 1385–1398 (2006) · Zbl 1102.92044 · doi:10.1016/j.amc.2006.05.025
[10] Yuan, S.L., Song, Y.L.: Stability and Hopf bifurcations in a delayed Leslie–Gower predator–prey system. J. Math. Anal. Appl. 355, 82–100 (2009) · Zbl 1170.34051 · doi:10.1016/j.jmaa.2009.01.052
[11] Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592–627 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[12] Nisbet, R.M., Gurney, W.S.C.: Modelling Fluctuating Populations. Wiley Interscience, New York (1982) · Zbl 0593.92013
[13] Saha, T., Bandyopadhyay, M.: Dynamical analysis of a delayed ratio-dependent prey–predator model within fluctuating environment. Appl. Math. Comput. 196, 458–478 (2008) · Zbl 1153.34051 · doi:10.1016/j.amc.2007.06.017
[14] Bandyopadhyay, M., Saha, T., Pal, R.: Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment. Nonlinear Anal. Hybrid Syst. 2, 958–970 (2008) · Zbl 1218.34098 · doi:10.1016/j.nahs.2008.04.001
[15] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Application of Hopf-bifurcation. Cambridge University Press, Cambridge (1981) · Zbl 0474.34002
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