Hybrid delayed feedback controller design in a nutrient-microorganism system accompanying time delay

During the past decades, delayed differential dynamical equation has displayed a great application in portraying the interplay of diverse chemical substances in chemistry. In the research, relying on the earlier publications, we propose a novel nutrient-microorganism system accompanying time delay. Exploiting fixed point theorem, inequality skills and a proper function, we acquire the parameter criteria on well-posedness of the solution of the established delayed nutrient-microorganism system. Utilizing Hopf bifurcation theory and stability criterion of delayed dynamical system, we deal with the onset of Hopf bifurcation phenomenon and stability trait of the established delayed nutrient-microorganism system. A set of novel delay-independent parameter conditions on bifurcation phenomenon and stability of the system are provided. Designing both different hybrid delayed feedback controllers, we have successfully controlled the delay length of occurrence of bifurcation phenomenon and stability domain of the system. The role of time delay in controlling bifurcation and stabilizing system is analyzed. Numerical experiment figures are given to support the gained primary outcomes. The acquired assertions of this research are completely innovative and can provide some important instructions in regulating and balancing the densities of microorganism and nutrient.