During this period of confinement I have just start studying on my own the book of J. D. Murray ''Mathematical Biology II: Spatial Models and Biomedical Applications''. Concretely I have read chapter II and III where he talks about spatial pattern formation with reaction diffusion systems and its applications. I am going to show what I have understand and the doubts that have arisen after the reading. I would be very grateful if someone could solve some of these doubts and tell me if I am following the correct way.
We start considering the reaction diffusion system
$$\begin{array}{l}u_{t}=\gamma f(u, v)+\nabla^{2} u, \quad v_{t}=\gamma g(u, v)+d \nabla^{2} v \\ \mathbf{n} \cdot \nabla u=0,\;\; \mathbf{n} \cdot \nabla v=0 \quad \mathbf{r} \text { on } \partial B ; \quad u(\mathbf{r}, 0), v(\mathbf{r}, 0) \text { given }\end{array}$$
Let $(u_0,v_0)$ the positive solution of $f(u,v)=g(u,v)=0$. Firstly we need to find conditions which ensure that this homogeneous steady state is stable to small perturbations in the absence
of diffusion but unstable to small spatial perturbations when diffusion is present. (Why is this necessary for a pattern to form?)
Time stablity (in absence of diffusion) is guaranteed under the hypothesis $tr(A)<0$ and $det(A)>0$, where $A=\left(\begin{array}{ll}
f_{u} & f_{v} \\
g_{u} & g_{v}
\end{array}\right)_{u_{0}, v_{0}}$.
For spatial instability we linearise the system about $(u_0,v_0)$ given rise to
$$\mathbf{w}_{t}=\gamma A \mathbf{w}+D \nabla^{2} \mathbf{w}, \quad D=\left(\begin{array}{ll}
1 & 0 \\
0 & d
\end{array}\right) \mathbf{w}=\left(\begin{array}{l}
u-u_{0} \\
v-v_{0}
\end{array}\right).$$
We make an ansatz of the form $\mathbf{w}=T(t)R(r)\mathbf{w_0}$ and we obtain$$\dfrac{T'}{T}\mathbf{w_0}=A \mathbf{w_0}+\dfrac{\Delta R}{R}\mathbf{w_0}$$ thus $T'(t)=\lambda T(t)$ and $\Delta R(r)=-k^2 \Delta R(r), \; n\cdot\nabla R =0$ in $\partial B$. (Why $k^2$ and not an arbitrary constant?)
This provides the condition: $(\lambda I_d -J+k^2D)w_0=0$. So, $\lambda(k^2)$ is a eigenvalue of the matrix $J-k^2D$ and we want that its real part is positive for at least one $k^2$. This occurs if $df_u+g_v>0$ and $\frac{1}{4d}(df_u+g_v)^2>det(A)$.
Let $k_1,k_2$ be the two integers such that $Re(\lambda(k^2))>0$ for every $k_1^2<k^2<k_2^2$ ($k_1$ and $k_2$ must be positive?), for large time we can approximate $\mathbf{w}$ by $\sum_{k_1}^{k_2} c_k e^{\lambda(k^2)t}R_k(r)$. (However, this solution is not bounded when $t\to\infty$, Murray comment this fact in the last paragraph of page 93 but I don't understand). Finally, we approximate $u\approx u_0+\sum_{k_1}^{k_2} c_k e^{\lambda(k^2)t}R_k(r)$. (How this solution give rise to a patterns?)
Sorry in advance if the quiestion is too long. Every help will be welcome.
Thanks.
Best Answer
First question: Why is this necessary for a pattern to form? Well, in general this is not necessary, but it is essential for the paradigm in which we work. We first assume the background is stable, and then, we perturb a parameter such a small band of wave-numbers becomes unstable. So there is no a priori reason this should word, but it gives the possibility to pattern formation, and it turned out to be a very effective possibility. Second question: Why $𝑘^2$ and not an arbitrary constant? That is purely a matter of convenience, as $k^2\geq0$ so you have control over the sign, and you'll see this is very common in the literature. Question three: $𝑘_1$ and $𝑘_2$ must be positive?. $\lambda$ is a funcion of $k^2$, so $\lambda$ is always symmetric in $k$, so you always get two unstable bands, positive and negative. Question four: How this solution give rise to a patterns? It is very important to note here that solutions to $T(t)R(r)$ are of the form $\exp(\lambda(k^2)t)\exp(ik^2r)$ for $K$ in the unstable band (Note: the negative unstable band gives rise to the complex conjugate of the positive band). Hence, the solutions that appear from the background are periodic functions (sines and cosines) with period $k^2$. These oscillatory functions are exactly the patterns we are looking for. The main follow up question is now: do these patterns blow up? Hopefully not, but to prove this you need a form of (weakly) nonlinear stability analysis. I.e, you need to prove that the the sines and cosines (with an appropriate constant in front) are approximate solutions to the full nonlinear problem. The two main techniques (I know at least) are center manifold reductions and modulation equations. A bit of googling should help you out here, a good start (written by physicists) is here https://openaccess.leidenuniv.nl/bitstream/handle/1887/5477/850_065.pdf?sequence=1.