A bacteriophage model based on CRISPR/Cas immune system in a chemostat

Mengshi Shu; Rui Fu; Wendi Wang

doi:10.3934/mbe.2017070

Article Contents

2017, Volume 14, Issue 5&6: 1361-1377. Doi: 10.3934/mbe.2017070

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A bacteriophage model based on CRISPR/Cas immune system in a chemostat

Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Wendi Wang
* Corresponding author: Wendi Wang

Received: May 30, 2016

Accepted: November 20, 2016

Published: November 2017

Supported in part by the NSF of China (11571284).

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Abstract

Clustered regularly interspaced short palindromic repeats (CRISPRs) along with Cas proteins are a widespread immune system across bacteria and archaea. In this paper, a mathematical model in a chemostat is proposed to investigate the effect of CRISPR/Cas on the bacteriophage dynamics. It is shown that the introduction of CRISPR/Cas can induce a backward bifurcation and transcritical bifurcation. Numerical simulations reveal the coexistence of a stable infection-free equilibrium with an infection equilibrium, or a stable infection-free equilibrium with a stable periodic solution.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 34D20.

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Figure 1. Bifurcation graphs for $\kappa\geq\mu$ in case $(C1)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.01$. Panel (b) indicates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents the branch point $E_1$

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Figure 2. Bifurcation graphs for case $(C2)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.3$ and panel (b) demonstrates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches

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Figure 3. Bifurcation graphs for case $(C3)$. Panel (a) shows the forward bifurcation at $E_1$ and a transcritical bifurcation at $E_3$ when $\varepsilon=0.1$ and panel (b) demonstrates the bistable phenomena between $E_1, ~E_3$ and a transcritical bifurcation at $E_4$ when $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches

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Figure 4. Graphs of bistable behaviors in case $(C1)$ with $\kappa\geq\mu$. Panel (a) shows the bistability of the infection-free equilibrium and an coexist equilibrium where $\varepsilon=0.01, ~b=130$. Panel (b) indicates the bistable coexistence of the infection-free equilibria with a stable periodic solution where $\varepsilon=0.9, ~b=260$

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