Mathematical analysis of fractional-order chemostat model with time delay - Mohd Aris, Nor Afiqah (2021)
The fractional-order differential equations (FDEs) is studied to describe the dynamic behaviour of a chemostat system. This study also investigated the FDEs with time delay to examine the effect of time delay on the behaviour of a chemostat system. The integer-order chemostat model in the form of the ordinary differential equation (ODEs) is extended to the FDEs. The stability and bifurcation analyses of the fractional-order chemostat model are discussed by using the Adams-type predictor-corrector method. Furthermore, the behaviour of the fractional-order chemostat system that considered time delay was also observed by using the modified Adams-type predictor-corrector method. The result shows that increasing or decreasing the value of the fractional order, α, may stabilise the unstable state of a chemostat system and also may destabilized the stable state of the chemostat system depend on the predefined parameter values. The increasing the value of the initial substrate concentration, S0 may destabilise the stable state of a chemostat system and stabilise the unstable state of the system. Therefore, the running state of a fractional-order chemostat system is affected by the value of α and the value of the initial substrate concentration, S0. In actual application, the value of the initial substrate should remain at S0 ≥ 2.54 to ensure that the chemostat system is unstable state. This is because there will be some change in amount of the cell mass concentration whether increase or decrease when the system is unstable, so chemostat system can be well controlled in order to be suitable for cell mass production. Other than that, the convergence speed of nearby trajectories increased when the value of α decreased. These results may be important to fastest the calculation time to achieve the steady-state in order to design the suitable state of the chemostat system. It is also observed that the inclusion of time delay can transform a stable state into a limit cycle and an unstable state with the appropriate choice of time delay value. Therefore, the suitable value of time delay can be chosen properly to ensure the dynamic behaviour of the chemostat system will always be unstable state and hence is suitable for the production of cell mass.