Stability Analysis¶

Let's return to the original equations. We've seen that for certain conditions, pattern formations occurs.

Unfortunately, solving the equations directly for non-linear reaction functions is often not possible. Instead we can look at what happens when the the system is perturbed slightly from equilibrium.

Linearising the Equations¶

We start by assuming there is some concentrations, $a_{0}$ and $b_{0}$, for which the system is stable. This means that

$R_{a}(a_{0},b_{0}) = 0$

$R_{b}(a_{0},b_{0}) = 0$

Around these solutions, we look at the time dependence of small perturbations around these values

$x = a - a_{0}$

$y = b - b_{0}$

And linearise the reaction equations

$R_{a}(a,b) \approx r_{aa}x + r_{ab}y$

$R_{b}(a,b) \approx r_{ba}x + r_{bb}y$

where $r_{ij} = \frac{\partial R_{i}}{\partial j}$.

These approximations give us a set of linear equations, written in vector form as

$\dot{\mathbf{x}} = D

abla^{2} \mathbf{x} + R \mathbf{x}$

Where $R = \left(\begin{matrix} r_{11} & r_{12} \\ r_{21} & r_{22}\end{matrix}\right)$ and $D = \left(\begin{matrix} D_{a} & 0 \\ 0 & D_{b}\end{matrix}\right)$

Fourier Transform¶

If we impose periodic boundary conditions to this equation, a natural solution can be found by applying a Fourier Transformation to $\mathbf{x}$. If we call the reciprocal coordinate $k$, and the Fourier transform of $\mathbf{x}$ as $\tilde{\mathbf{x}}$, then the transformed equation is

$\dot{\tilde{\mathbf{x}}} = (R - k^{2}D) \tilde{\mathbf{x}}$

Which, has solutions of the form

$\tilde{\mathbf{x}}(t) = \tilde{\mathbf{x}}(0) e^{\omega t}$

To find $\omega$ we plug this solution back into our transformed equation to get

$\omega \tilde{\mathbf{x}} = (R - k^{2}D) \tilde{\mathbf{x}}$

Showing that $\omega$ is just the eigenvalue of $(R - k^{2}D)$.

We now have an equation for the time dependence of our system in the Fourier domain. Using it, we can now discuss what we mean by stability. Our system is considered stable if small perturbations around the homogeneous do not cause it to move further away from the stable solutions.

In terms of our solution, $\tilde{\mathbf{x}}(0) e^{\omega t}$, stability means that the values of $\omega$ does not have positive real parts for all values of $k$. If $\omega$ is negative, the perturbation will decay away. If $\omega$ is imaginary, it will oscillate around the stable state. However, if it is positive and real any small perturbation will grow exponentially, until a high order term of the reaction equation becomes important.

To find $\omega$, we need to solve the equation

$\hbox{Det}(R - k^{2}D - \omega I) = 0 $

Writing $J = R - k^{2}D$, this equitation takes the form

$\omega^{2} - \omega\hbox{Tr}(J) + \hbox{Det}(J) = 0$

Solving for $\omega$, we get

$\omega = \frac{1}{2}(\hbox{Tr}(J) \pm \sqrt{\hbox{Tr}(J)^{2} - 4 \hbox{Det}(J) })$

Conditions for (in)Stability¶

From our initial assumption that there was a stable homogeneous state, we require that $\omega$ has negative real parts where $k = 0$, which corresponds to the spatially homogeneous solution. For this to be true, we require that

$\hbox{Tr}(R) < 0$

$\hbox{det}(R) > 0$

Or, in terms of the components of $R$:

$r_{aa} + r_{bb} < 0$

$r_{aa}r_{bb} - r_{ab}r_{ba} > 0$

For an instability to now occur at finite wavelength, we need one of the following conditions to hold:

$\hbox{Tr}(J) > 0$

$\hbox{det}(J) < 0$

Because $\hbox{Tr}(J) = \hbox{Tr}(R) - k^{2}(d_{a} + d_{b})$, the first condition cannot hold for any real $k$. This means the we require the second to hold, or, after a bit of algebra

$k^{4}d_{a}d_{b} - k^{2}(d_{a}r_{bb} + d_{b}r_{aa}) + (r_{aa}r_{bb} - r_{ab}r_{ba}) < 0$

for some real value of $k$. Once again, we need to solve a quadratic equation. To do this we note that because $k$ needs to be real, $k^{2}$ must be positive. This means that at least one root of quadratic equation in $k^{2}$ needs to be positive and real. This condition is only met when

$d_{b}r_{aa} + d_{a}r_{bb} > 2\sqrt{d_{a}d_{b}(r_{aa}r_{bb} - r_{ab}r_{ba})} > 0$

And that's it. We've derived that conditions for the diffusion-reaction equations to be stable to small perturbations.

We can write the complete requirements as