Introduction to the Modeling and Analysis of Complex Systems

14.4. LINEAR STABILITY ANALYSIS OF REACTION-DIFFUSION SYSTEMS 287Here I used S = sin(ωx + φ) only in the expressions above to shorten them.equations can be summarized in a single vector form about ∆f,sin(ωx + φ) ∂∆f∂tThese= R(f eq + sin(ωx + φ)∆f) − Dω 2 sin(ωx + φ)∆f, (14.90)where R is a vector function that represents all the reaction terms, and D is a diagonalmatrix whose diagonal components are D i for the i-th position. Now that all the diffusionterms have been simplified, if we can also linearize the reaction terms, we can completethe linearization task. And this is where the Jacobian matrix is brought back into thespotlight. The reaction terms are all local without any spatial operators involved, andtherefore, from the discussion in Section 5.7, we know that the vector function R(f eq +sin(ωx + φ)∆f) can be linearly approximated as follows:⎛⎞∂R 1∂R 1 ∣R(f eq + sin(ωx + φ)∆f) ≈ R(f eq ) +⎜⎝∂R 1∂f 1∂R 2 ∂R 2∂f 1.∂R n∂f 1⎛= sin(ωx + φ)⎜⎝∂f 2. . .∂f n∂R∂f 2. . . 2∂f n.. .. .∂R n ∂R∂f 2. . . n∂f n∂R 1 ∂R 1∂f 1∂R 2 ∂R 2∂f 1.∂R n∂f 1⎟sin(ωx + φ)∆f⎠∣f=feq∂R 1∂f n∂R 2∂f n∂f 2. . .∂f 2. . ... . . .∂R n ∂R∂f 2. . . n∂f n(14.91)⎞⎟∆f (14.92)⎠∣f=feqNote that we can eliminate R(f eq ) because of Eqs. (14.83)–(14.85). By plugging this resultinto Eq. (14.90), we obtainsin(ωx + φ) ∂∆f = sin(ωx + φ)J| f=feq ∆f − Dω 2 sin(ωx + φ)∆f,∂t(14.93)∂∆f= ( J − Dω 2) | f=feq ∆f,∂t(14.94)where J is the Jacobian matrix of the reaction terms (R). Very simple! Now we just needto calculate the eigenvalues of this coefficient matrix to study the stability of the system.The stability of a reaction-diffusion system at its homogeneous equilibrium state f eqcan be studied by calculating the eigenvalues of(J − Dω2 ) | f=feq , (14.95)