Analysis And Simulation Of Nonlinear Behavior In The Model Of Microbial Continuous Cultivation

Nonlinearity abroad exists in the nature, and it is also one of the influential tasks in basic studies of natural science. The biochemical reactions include abundant nonlinear phenomena because of the complicacy of cell's interior structure. With the model of producing 1,3-propanediol by microbial continuous cultivation (abbreviated as MMCC here) as example, we study the effects of delay on the nonlinear behaviors in the model of microbial continuous cultivation, which has the character of inhibition of substrate and product and metabolic overflow. MMCC qualitatively describes the multiplicity observed in experiments, but failed to produce oscillations. In this article, a delay is introduced into MMCC, and the revised model exihibits oscillations. The main contributions of this paper are as follows:1 In the second chapter, the curves of stationary bifurcation values and Hopf bifurcation values of the MMCC are obtained by applying bifurcation theory and numerical method. The region where multiplicity exists is given on the basis of eigenvalues. The direction and stability of Hopf bifurcation are discussed.2 In the third chapter, a discrete delay is introduced into MMCC, and the effects of delay on local stability of positive equlibria are studied. The region where Hopf bifurcation exists is given, and the direction and stability of Hopf bifurcation are determined by using the normal form theory and center manifold theorem. By continuation of periodic solutions, it is observed that the bifurcating periodic solutions, which emerge from Hopf bifurcation, lose their stability and period doubling bifurcation occurs when a delay goes beyond a certain value. Numerical calculations suggest that chaos would occur with increasing the delay. These results qualitatively describe the oscillations exhibited in experiments.3 In the fourth chapter, a continuous delay is introduced into MMCC, and the effects of mean delay on local stability of positive equlibria are discussed. The effects of continuous delay and discrete delays on period and position of periodical solutionsof MMCC are compared.4 In the last chapter, dynamical specific growth rate, dynamical specific consumption rate and dynamical formation rate are introduced into the MMCC with three sorts of product, and then the revised model is neutral delay differential equations. To make the model describe oscillation observed in experiment better, parameter identification problem is introduced to solve the uncertain parameters. The results quantitatively describe oscillations observed in experiment.