Modelling of the complex nonlinear processes: Determination of the instability region by the stoichiometric network analysis

Abstract

Modeling of a complex nonlinear process whatever it describes is a serious task from mathematical point of view. Generally, in a complex nonlinear reaction system there is possibility to find region in parameter space where the main steady state is unstable. In that region numerous self-organized dynamic states, such as multistability, periodicity and chaos can be established. Although mentioned dissipative structures are common in nature (many biochemical processes are in the oscillatory states), the region of parameters where they appear is often very narrow. Therefore, mathematical modeling of the process under consideration, with clear and precise determination of instability region is desirable. If the model can be reduced to two or three variables, the locus of unstable steady (nonequilibrium stationary) states can be easily obtained. Models with more variables must be explored by some general method, such as the stoichiometric network analysis, a known powerful method for the examination of complex reaction systems, the possible pathways in them, and corresponding stability analysis. However, in the form proposed by B. Clarke, the general analytical expression for the instability condition related to experimental information was not achieved, although geometrical solutions of the problem were suggested. In last papers, we have offered the procedure for obtaining the instability condition in the function of reaction rates. Our aim here is to present the mathematical derivation of instability condition with application of theory to selected models.