Linear Analysis of Turing Instability in Systems of Chemical Kinetics

The goal of the work is to find the region of diffusion instability on the plane of parameters of the Schnackenberg and Brusselator systems, as well as for a system that generalizes these systems. The Schnakenberg system is considered in two cases: under the assumption that the equilibrium position is positive and under the additional (stronger) assumption that the parameters of the system are positive. To study systems using the methods of stability theory, a transition to generalized variables is used, in which the Jacobian matrices of the indicated systems in the vicinity of equilibrium positions coincide. It is shown that in the first case, in the absence of additional restrictions, the regions of necessary conditions for Turing instability are not empty and are not limited in the plane of system parameters. In the second case, under stronger assumptions, it is shown that the range of necessary conditions is limited. Conditions have been obtained under which the area of necessary conditions can be empty. Using the transition to generalized variables, analytical expressions for the boundaries of the domain of sufficient conditions are obtained in the presence of additional assumptions corresponding to the positivity of the parameters for the Schnakenberg system. The case of large diffusion coefficients is considered, when the minimum critical wave number is equal to one, as well as cases in which the minimum wave number is greater than one.