Morbidostat 是一個通過不斷提高抗生素濃度來研究細菌演化進程的細菌培養裝置。在本文中，我們首先設置了一個合理的假設條件，並在該假設條件下證明了藥物注射系統的解的有界性。 隨後，我們分別分析了在三種不同突變情況下系統的局部和全域穩定性。最後，我們總結了當閾值參數U發生改變時，包括原生細菌和突變體在內的所有物種的生存狀況。

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration and maintains a constant challenge for study of evolutionary pathway \cite{5}. In this work, we first change the hypothesis of \cite(5) in a more reasonable way, and prove the boundedness of the solutions in drug on-drug off system. Then we analyze the local stability and global stability of the system with no mutation, only forward mutation and forward-backward mutation respectively. We conclude the survival condition of all species including the wide type and the mutants when the threshold parameter U varies.

1.Introduction and models . . . . . . . . . . .1
1.1.Introduction. .. . . . . . . . . . . . . . .1
1.2.Description of the models. . . . . . . . . . 2
2.Preliminary . . . . . . . . . . .4
3.Dynamics of drug on-drug o model with no mutations . . . . . . 4
3.1.The behavior of solutions of the system (3.1) . . . . . . .4
3.2.The results of the asymptotic behavior of the solutions of
model with no mutations . . . . . . . . . . . . . . . . . . 7
4.Dynamics of drug on-drug o model with only forward mutations. . . . . . . . . . .11
4.1.The behavior of the solutions of the system (4.1) . . . . . 11
4.2.The results of the asymptotic behavior of the solutions of model with only forward mutations . . . . . . .. . . . . .21
5.Dynamics of drug on-drug o model with forward-backward mutations
. . . . . . . . . . .23
6.Numerical simulations. . . . . . . . . . .25
7.References. . . . . . . . . . .27

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