Abstract
In this paper, a stochastic competition model with time delay and harvesting is investigated. By means of the stochastic center manifold reduction principle and stochastic averaging method, the model is simplified to a one-dimensional Markov diffusing process. The singular boundary theory and invariant measure are applied in analyzing the stochastic stability and bifurcation. The T-S fuzzy model of the system is constructed, and the \(H_{\infty }\) state feedback controller is designed to eliminate the instability phenomenon by using a linear matrix inequality approach. Finally, numerical simulations are given to demonstrate our results.
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References
Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139–153 (1990)
Chakraborty, K., Chakraborty, M., Kar, T.K.: Optimal control of harvest and bifurcation of a prey-predator model with stage structure. Appl. Math. Comput. 217, 8778–8792 (2011)
Chen, F.: Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control. Appl. Math. Comput. 170, 1452–1468 (2005)
Chen, B., Liu, X.: Delay-dependent robust control for T-S fuzzy systems with time delay. IEEE Trans. Fuzzy Syst. 13, 544–556 (2005)
Cushing J. M.: An introduction to structured population dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia(1998)
Ehsan, B., Yousef, A., Daryoush, A., Jafar, M.A., Matjaž, P.: Control of dynamics via identical time-lagged stochastic inputs. Chaos 30, 013143 (2020)
Fofana, M.S.: Asymptotic stability of a stochastic delay equation. Probab. Eng. Mech. 17, 385–392 (2002)
Gao, Y., Tian, S.: Dynamics of a stochastic three-species competitive model with \(\rm L\grave{e}vy\) jumps. Int. J. Biomath. 11, 1850075 (2018)
Han, X., Ma, Y.: Sampled-data robust \(H_{\infty }\) control for T-S fuzzy time-delay systems with state quantization. Int. J. Control Autom. Syst. 17, 46–56 (2019)
Hou, Z.: Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems. Appl. Math. Lett. 25, 195–199 (2012)
Hu, D., Li, Y., Liu, M., Bai, Y.: Stability and Hopf bifurcation for a delayed predator-prey model with stage structure for prey and Ivlev-type functional response. Nonlinear Dyn. 99, 3323–3350 (2020)
Huang Z.: The Study of dynamic properties of stochastic differential systems with delay. South China University of Technology(2011)
Huang, D., Wang, H., Feng, J., Zhu, Z.: Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics. Chaos Solitons Fractals 27, 1072–1079 (2006)
Huang, Z., Yang, Q., Cao, J.: Complex dynamics in a stochastic internal HIV model. Chaos Solitons Fractals 44, 954–963 (2011)
Huang, X., Chen, F., Xie, X., Zhao, L.: Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting. Open Math. 17, 856–873 (2019)
Knobloch, E., Wiesenfeld, K.A.: Bifurcations in fluctuating systems: the center-manifold approach. J. Stat. Phys. 33, 611–637 (1983)
Kon, R., Saito, Y., Takeuchi, Y.: Permanence of single-species stage-structured models. J. Math. Biol. 48, 515–528 (2004)
Liu, M., Qiu, H., Wang, K.: A remark on a stochastic predator-prey system with time delays. Appl. Math. Lett. 26, 318–323 (2013)
Lu, C., Ding, X.: Dynamical behavior of stochastic delay Lotka-Volterra competitive model with general \(\rm L\grave{e}vy\) jumps. Physica A 531, 121730 (2019)
Manivannan, R., Samidurai, R., Cao, J., Matjaž, P.: Design of resilient reliable dissipativity control for systems with actuator faults and probabilistic time-delay signals via sampled-data approach. IEEE Trans. Syst, Man. Cyber. Syst. 50, 4243–4255 (2020)
Neubert, M.G., Caswell, H.: Density-dependent vital rates and their population dynamic. J. Math. Biol. 41, 103–121 (2000)
Qiu, H., Deng, W., Xiang, M.: Optimal harvesting strategies of a stochastic competitive model with S-type distributed time delays and \(\rm L\grave{e}vy\) jumps. Boundary Value Problems 2021, 1–17 (2021)
Rudnicki, R.: Long-time behaviour of a stochastic prey-predator model. Stochastic Process. Their Appl. 108, 93–107 (2003)
Song, X., Chen, L.: Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170, 173–186 (2001)
Sun, Y., Xu, J., Chen, C., Lin, G., Hsieh, W.H.: Fuzzy \(H_{\infty }\) robust control for magnetic levitation system of maglev vehicles based on T-S fuzzy model: Design and experiments. J. Intell. Fuzzy Syst. 36, 1–12 (2018)
Wu, B., Chang, X., Zhao, X.: Fuzzy \(H_{\infty }\) output feedback control for nonlinear NCSs with quantization and stochastic communication protocol. IEEE Trans. Fuzzy Syst. 26, 1–11 (2020)
Xing S. Y.. Analysis and control of a class of singular stochastic systems with Itô-type. Northeastern University(2015)
Yan, H.C., Wang, T.T., Zhang, H., Shi, H.B.: Event-triggered \(H_{\infty }\) control for uncertain networked TCS fuzzy systems with time delay. Neurocomputing 157, 273–279 (2015)
Zeeman, M.L.: Extinction in competitive Lotka-Volterra systems. Proc. Am. Math. Soc. 123, 87–96 (1995)
Zhang, Y., Zhang, Q.: Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting. Nonlinear Dyn. 66, 231–245 (2011)
Zhang, Y., Zhang, Q., Zhang, T.: \(H_{\infty }\) control of generalized bio-economic systems. J. Northeastern Univ. (Natural Science) 32, 1369–1373 (2011)
Zhu B.: Analysis and control for a kind of T-S fuzzy descriptor dystem. Northeastern University(2006)
Zhu, C., Yin, G.: On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 357, 154–170 (2009)
Funding
This study was funded by the National Natural Science Foundation of China (61703083 and 61673100), China Scholarship Council (201706085041) and Fundamental Research Funds for the Central Universities (N2104007).
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Zhang, Y., Zhang, J. & Liu, X. Bifurcation analysis and \(\pmb {H_{\infty }}\) control of a stochastic competition model with time delay and harvesting. Nonlinear Dyn 109, 1217–1232 (2022). https://doi.org/10.1007/s11071-022-07381-y
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DOI: https://doi.org/10.1007/s11071-022-07381-y