Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

References

• Chen, L., and Chen, J. 1993. Nonlinear Biological Dynamic System. Monmouth Junction, NJ: Science Press.
   Google Scholar
• Waltman, P., and Smith, H. L. 1995. The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Canale, R. P. 1970. An analysis of models describing predator-prey interaction. Biotechnology & Bioengineering 12(3):353–378.
   Web of Science ®Google Scholar
• Kuang, Y. 2002. Limit cycles in a chemostat-related model. SIAM Journal on Applied Mathematics 49(6):1759–1767.
   Web of Science ®Google Scholar
• Li, B., and Kuang, Y. 2000. Simple food chain in a chemostat with distinct removal rates. Journal of Mathematical Analysis & Applications 242(1):75–92.
   Web of Science ®Google Scholar
• Ali, E., Asif, M., and Ajbar, A. H. 2013. Study of chaotic behavior in predator-prey interactions in a chemostat. Ecological Modelling 259(259):10–15.
   Web of Science ®Google Scholar
• Wang, F., Pang, G., and Zhang, S. 2009. Analysis of a Lotka–Volterra food chain chemostat with converting time delays. Chaos Solitons & Fractals 42(5):2786–2795.
   Web of Science ®Google Scholar
• Liu, J., and Zheng, S. 2002. A reaction-diffusion system arising from food-chain in an unstirred chemostat. Journal of Biomathematics 17(3):263–272.
   Google Scholar
• Wang, F., Zhang, S., Chen, L., and Sun, L. 2006. Bifurcation and complexity of monod type predator-prey system in a pulsed chemostat. Chaos Solitons & Fractals 27(2):447–458.
   Web of Science ®Google Scholar
• Xu, C., Yuan, S., and Zhang, T. 2013. Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate. Abstract & Applied Analysis 2013(1):233–242.
   Google Scholar
• Mao, X. 2008. Stochastic Differential Equations and Applications. Chichester, UK: Horwood.
   Google Scholar
• Wang, K. 2010. Stochastic Biological Mathematical Modelling. Monmouth Junction, NJ: Science Press.
   Google Scholar
• Ji, C. 2011. The asymptotic behavior of stochastic biological models and epidemic models. PhD thesis, Northeast Normal University.
   Google Scholar
• Liu, M., and Bai, C. 2016. Analysis of a stochastic tri-trophic food-chain model with harvesting. Journal of Mathematical Biology 73(3):597–625.
   PubMed Web of Science ®Google Scholar
• Dong, Q. 2014. Asymptotic behavior of a stochastic ratio-dependent chemostat model. Journal of Shandong University 49(3): 68–72, 78.
   Google Scholar
• Imhof, L., and Walcher, S. 2005. Exclusion and persistence in deterministic and stochastic chemostat models. Journal of Differential Equations 217(1):26–53.
   Web of Science ®Google Scholar
• Grasman, J., De Gee, M., and Van Herwaarden, O. A. 2005. Breakdown of a chemostat exposed to stochastic noise. Journal of Engineering Mathematics 53(3):291–300.
   Web of Science ®Google Scholar
• Chen, Z., and Zhang, T. 2013. Dynamics of a stochastic model for continuous flow bioreactor with contois growth rate. Journal of Mathematical Chemistry 51(3):1076–1091.
   Web of Science ®Google Scholar
• Chen, Z., and Zhang, T. 2013. Long time behaviour of a stochastic model for continuous flow bioreactor. Journal of Mathematical Chemistry 51(2):451–464.
   Web of Science ®Google Scholar
• Collet, P., Martinez, S., Meleard, S., and San Martin, J. 2013. Stochastic models for a chemostat and long time behavior. Advances in Applied Probability 45(3):822–836.
   Web of Science ®Google Scholar
• Campillo, F., Joannides, M., and Larramendy-Valverde, I. 2011. Approximation of the Fokker–Planck equation of the stochastic chemostat. Mathematics & Computers in Simulation 99(1):37–53.
   Google Scholar
• Xu, C., and Yuan, S. 2015. An analogue of break-even concentration in a simple stochastic chemostat model. Applied Mathematics Letters 48:62–68.
   Web of Science ®Google Scholar
• Zhao, D., and Yuan, S. 2016. Critical result on the break-even concentration in a single-species stochastic chemostat model. Journal of Mathematical Analysis and Applications 434:1336–1345.
   Web of Science ®Google Scholar
• Liu, M., and Fan, M. 2016. Permanence of stochastic Lotka–Volterra systems. Journal of Nonlinear Science 27(2):425–452.
   Google Scholar
• Lahrouz, A., Settati, A., and Akharif, A. 2017. Effects of stochastic perturbation on the sis epidemic system. Journal of Mathematical Biology 74:469–498.
   PubMed Web of Science ®Google Scholar
• Evans, S. N., Ralph, P. L., Schreiber, S. J., and Sen, A. 2013. Stochastic population growth in spatially heterogeneous environments. Journal of Mathematical Biology 66(3):423–469.
   PubMed Web of Science ®Google Scholar
• Durrett, R. 1996. Stochastic Calculus: A Practical Introduction. Boca Raton, FL: CRC Press.
   Google Scholar
• Khasminskii, R. 1980. Stochastic Stability of Differential Equations. Alphen aan den Rijn, the Netherlands: Sijthoff & Noordhoff.
   Google Scholar
• Liu, M., and Bai, C. 2016. Optimal harvesting of a stochastic mutualism model with Lévy jumps. Applied Mathematics & Computation 276:301–309.
   Web of Science ®Google Scholar
• Liu, M., and Fan, M. 2017. Stability in distribution of a three-species stochastic cascade predator-prey system with time delay. IMA Journal of Applied Mathematics. http://dx.doi.org/10.1093/imamat/hxw057
   Google Scholar
• Dang, H. N., and Yin, G. 2017. Coexistence and exclusion of stochastic competitive Lotka–Volterra models. Journal of Differential Equations 262(3):1192–1225.
   Web of Science ®Google Scholar
• Higham, D. J. 2001. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review 43(3):525–546.
   Web of Science ®Google Scholar
• Hu, G., and Wang, K. 2011. The estimation of probability distribution of sde by only one sample trajectory. Computers & Mathematics With Applications 62(4):1798–1806.
   Web of Science ®Google Scholar
• Liu, M., and Wang, K. 2013. Dynamics of a two-prey one-predator system in random environments. Journal of Nonlinear Science 23(5):751–775.
   Web of Science ®Google Scholar