A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient

Brittni Hall; Xiaoying Han; Peter E. Kloeden; Hans-Werner van Wyk

doi:10.3934/dcdss.2022075

Article Contents

2022, Volume 15, Issue 10: 2889-2908. Doi: 10.3934/dcdss.2022075

This issue Previous Article Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations Next Article On a global climate model with non-monotone multivalued coalbedo

A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient

221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

^* Corresponding author: Hans-Werner van Wyk
^* Corresponding author: Hans-Werner van Wyk

Dedicated to Professor Georg Hetzer on the occasion of his 75th birthday

Received: August 2021

Early access: March 2022

Published: October 2022

This work is partially supported by FEDER-Junta de Andalucia, Spain (project P18-FR-4509).

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Abstract

A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results.

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Mathematics Subject Classification: Primary: 34D45, 35B41, 37B55, 92B05, 92D25; Secondary: 35B35, 45M10, 97M60.

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Figure 1. Plot of the fraction $ R(t) $ of bacteria in the gut for different initial conditions

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Figure 2. Plot showing the joint dynamics of the nutrient $ x(t) $ and the total bacteria $ Y(t) $ over the time interval $ [0,2] $ under assumptions (A1) and (A2). The arrows indicate the trajectory directions

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Figure 3. The time evolution of the fraction $ R(t) $ of bacteria in the gut and the total bacteria $ Y(t) $ under nonautonomous, periodic forcing with parameters satisfying Assumptions (A1), (A2), and (A3), but not (A4)

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Figure 4. Plots showing the convergence of the total bacteria $ Y(t) $ to $ 0 $ and that of the nutrition level $ x(t) $ to $ x^*(t) $ under nonautonomous, periodic forcing and Assumptions (A1)–(A4)

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Figure 5. Joint trajectories of the nutrition $ x $, the bacteria in the gut $ y_1 $, and the bacteria on the wall $ y_2 $ subject to nonautonomous, periodic forcing, with parameters safisfying (A1)–(A4)

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Figure 6. Plot of the joint dynamics of the nutrition level $ x $ and the total bacteria population $ Y $ under nonautonomous, periodic forcing, with parameters satisfying Assumptions (A1)–(A3), (A5), and (A6)

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Figure 7. Plot of the trajectories in Figure 6 on a logarithmic scale

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Figure 8. Trajectory plot $ (x(t),Y(t)) $ in the time interval $ [0,50] $ of the nutrition $ x $ and the total bacteria population $ Y $ under constant nutrient injection for various initial conditions

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Figure 9. Plot of the fraction of bacteria in the gut over time under autonomous forcing and Assumptions (A1)–(A3), (A5), and (A6)

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Table 1. Model parameters

Parameter	Meaning
$ D(t)> 0 $	input and output flow rate
$ I(t)> 0 $	input concentration of the nutrient
$ \mu> 0 $	ratio between output and input flow rates
$ a> 0 $	maximum consumption rate of the nutrient by the microorganisms
$ b \geq 0 $	growth rate of the microorganisms due to consumption
$ m> 0 $	half-saturation rate of the consumption function
$ \delta \in (0, 1) $	recycle rate from dead microorganisms to new microbe biomass
$ \nu \geq 0 $	collective death rate of microorganisms
$ r_1 \geq 0 $	rate at which microbe attaches to the wall
$ r_2 \geq 0 $	rate at which microbe detaches from wall
$ \alpha \geq 0 $	intra-specific competition rate of microbe population in reservoir
$ \gamma \geq 0 $	intra-specific competition rate of wall population of microbe

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