Dynamics of drug on-drug off models with mutations in morbidostat — Dedicated to the seventieth birthday of Professor Gail Wolkowicz

Ziran Cheng; Sze-Bi Hsu; Ziran Cheng; Sze-Bi Hsu

doi:10.3934/math.20231061

AIMS Mathematics

2023, Volume 8, Issue 9: 20815-20840. doi: 10.3934/math.20231061

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Dynamics of drug on-drug off models with mutations in morbidostat — Dedicated to the seventieth birthday of Professor Gail Wolkowicz

Ziran Cheng ¹,
Sze-Bi Hsu ^{2
,
,}

1.
Shen Zhen senior high school, Shenzhen, Guan-Dong, China
2.
Department of Mathematics, National Tsing-Hua University, Taiwan

Received: 04 October 2022 Revised: 03 June 2023 Accepted: 20 June 2023 Published: 29 June 2023
MSC : 34C, 92B, 94D

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration. It is used to study the evolutionary pathway. In this article, we construct mathematical models for the morbidostat. First we consider the case of no mutations, we study limiting systems and obtain criteria for the large time behavior of the solutions. From the theoretical results and numerical simulations, we conclude that there are two competitive exclusion states of either wild type or mutant type as the threshold parameter $ U $ varies. There are three cases, wild type bacteria excludes all mutants; a mutant dominates in the competition; oscillation between the above two states.

Next we study the systems of forward mutations and forward-backward mutations. Then we apply a result of pertubation for globally stable state.
- morbidostat,
- chemostat,
- wild type,
- mutants,
- drug inhibition,
- exclusion principle,
- coexistence,
- differential inequalities,
- impulse systems
Citation: Ziran Cheng, Sze-Bi Hsu. Dynamics of drug on-drug off models with mutations in morbidostat — Dedicated to the seventieth birthday of Professor Gail Wolkowicz[J]. AIMS Mathematics, 2023, 8(9): 20815-20840. doi: 10.3934/math.20231061

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Abstract

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration. It is used to study the evolutionary pathway. In this article, we construct mathematical models for the morbidostat. First we consider the case of no mutations, we study limiting systems and obtain criteria for the large time behavior of the solutions. From the theoretical results and numerical simulations, we conclude that there are two competitive exclusion states of either wild type or mutant type as the threshold parameter $ U $ varies. There are three cases, wild type bacteria excludes all mutants; a mutant dominates in the competition; oscillation between the above two states.

Next we study the systems of forward mutations and forward-backward mutations. Then we apply a result of pertubation for globally stable state.

References

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