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**Stability results for a class of differential equation and application in medicine.**
*(English)*
Zbl 1173.34331

Summary: A chemostat system incorporating hepatocellular carcinomas is discussed. The model generalizes the classical chemostat model, and it assumes that the chemostat is an increasing function of the concentration. The asymptotic behavior of solutions is determined. Sufficient conditions for the local and global asymptotic stability of equilibrium and numerical simulation are obtained, which is used to select the disease control tactics.

### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

92C50 | Medical applications (general) |

34D20 | Stability of solutions to ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

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\textit{Q. Zhan} et al., Abstr. Appl. Anal. 2009, Article ID 187021, 8 p. (2009; Zbl 1173.34331)

### References:

[1] | L. S. Chen, X. Song, and Z. Lu, Mathematical Models and Methods in Ecology, Sichuan Science and Techlogy, Chendu, China, 2003. |

[2] | Z. Yixing, “Differential equation applied to research of quantitative analysis in biomedicine,” Journal of Southwest University for Nationalities, vol. 37, no. 2, pp. 231-233, 2001. |

[3] | Z. Jiakun, Z. Lingli, and Z. Wei, “Epidemic model of hepatitis B without vaccination,” Journal of Xuzhou Normal University, vol. 20, no. 4, pp. 7-11, 2002. · Zbl 1037.92036 |

[4] | Q. Zhan, X. Xie, C. Wu, and S. Qiu, “Hopf bifurcation and uniqueness of limit cycle for a class of quartic system,” Applied Mathematics: A Journal of Chinese Universities, vol. 22, no. 4, pp. 388-392, 2007. · Zbl 1150.34402 |

[5] | X. Yang, Qualitative Theory of Dynamical Systems, Cheung Shing, Hong Kong, 2004. |

[6] | S.-B. Hsu and T.-W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763-783, 1995. · Zbl 0832.34035 |

[7] | G. Pang and L. Chen, “Global stability of chemostat models with ratio-dependent increase rate,” Journal of Guangxi Normal Universiy, vol. 24, no. 1, pp. 37-40, 2006. · Zbl 1100.34523 |

[8] | X. Xie and F. Chen, “Bifurcation of limit cycles for a class of cubic systems with two imaginary invariant lines,” Acta Mathematica Scientia. Series A, vol. 25, no. 4, pp. 538-545, 2005. · Zbl 1110.34314 |

[9] | F. Chen, “The permanence and global attractivity of Lotka-Volterra competition system with feedback controls,” Nonlinear Analysis: Real World Applications, vol. 7, no. 1, pp. 133-143, 2006. · Zbl 1103.34038 |

[10] | S. S. Pilyugin and P. Waltman, “Multiple limit cycles in the chemostat with variable yield,” Mathematical Biosciences, vol. 182, no. 2, pp. 151-166, 2003. · Zbl 1012.92044 |

[11] | Z. Zhang, T. Ding, W. Huang, and Z. Dong, Qualitative Theory of Differential Equations, Scientific, Beijing, China, 1985. |

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