Turing instability and Hopf bifurcation of a spatially discretized reaction-diffusion equations with two components under discretized boundary conditions

A spatially discretized reaction-diffusion equations with two components are considered. The equations are derived by using 3-point centered difference approximations for the second derivative. By considering the spatially discretized counterparts of periodic, Dirichlet and homogeneous Neumann boundary conditions for the reaction-diffusion equations, 17 kinds of spatially discretized boundary conditions are further given. Then the local stability of the spatially homogeneous steady state of the spatially discretized reaction-diffusion equations subject to one of the spatially discretized boundary conditions is studied by analyzing the linearized equations with the aid of decoupling method. Besides, the specific expressions of the spatial eigenvalues and eigenvectors related to the spatially discretized boundary conditions are given. Hence the occurrence conditions of Turing instability and Hopf bifurcation at this steady state are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are derived according to Hopf bifurcation theorem. Finally, an example is given to demonstrate the results both theoretically and numerically. The methods and computations can be extended to consider the spatially discretized reaction-diffusion equations with arbitrarily many components.