Exploring the Dynamics of Simple Inhibition Systems in Continuous Stirred-Tank Reactor: Mathematical Modelling and Bifurcation Analysis


  • Afifi Md Desa ᵃSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia; ᵇInstitute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia
  • Mohd Hafiz Mohd School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
  • Mohamad Hekarl Uzir School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia




Enzyme inhibitions, dynamical systems, bifurcation, phase plane analysis


The simple enzyme inhibition systems consist of competitive inhibition, noncompetitive inhibition and uncompetitive inhibition. In this work, we incorporated these simple inhibition systems in the continuous stirred-tank reactor (CSTR) and analysed the models using some techniques from dynamical systems and bifurcation analysis. Our aim is to investigate the behaviours of such systems and compare their overall dynamics. The phase portrait is constructed to simulate possible behaviours such as stable steady states, stable limit cycle, bistability between the steady state and the stable limit cycle and bistability between two steady states. The systems undergo bifurcational changes in dynamics as enzyme concentration, dilution rate and proportional control constant are varied. Moreover, we conducted a codimension two bifurcation analysis to examine the joint effects of dilution rate and proportional control constant on the systems behaviours. Our results revealed distinct dynamics for each inhibition system. Increasing the dilution rate led to a transition from low to high substrate concentrations, with competitive inhibition showing the highest tipping (or bifurcation) point where dynamical regimes change due to intense substrate-inhibitor competition. Elevating enzyme concentration reduced substrate concentration, particularly in non-competitive inhibition systems due to higher conversion rates. Furthermore, the proportional control constant had varying effects depending on the specific inhibition system. These findings emphasize the on the combined influences of distinct chemical procoesses in controlling reactor heat and optimizing bioprocess efficiency, considering the unique characteristics of each inhibition system. Overall, the dynamical study on these simple inhibition systems enables us to improve our understanding on the chemical processes involving enzymes with multiple types of inhibitors and may give some insights in its controlling process.


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