Mathematical problem definition¶
We generally define reaction-diffusion systems as:
for time \(t\in[0, T]\) and space
\(\symbfit x = [x, y, z]^{\text{T}} \in\heartsuit\subset\mathbb R^3\)
in the domain \(\heartsuit\), with initial conditions
\(\underline{u}(0, \symbfit x)\) and no-flux boundary conditions
\(0 = \symbfit{n}\cdot \symbfit D \nabla u\) for diffused variables on
\(\symbfit x \in \partial\heartsuit\). At each point in time and space,
the states vector
\(\underline{u}(t, \symbfit x) \in \mathbb R^{\mathtt{Nv}}\) consists of
Nv elements, the so-called states or variables, \(u_0\),
\(u_1\), …, \(u_{\mathtt{Nv}-1}\). For the diffusion term, we
define the diffusivity matrix \(\symbfit D(\symbfit x) \in
\mathbb R^{3\times 3}\) and the selection matrix
\(\underline{P} \in \mathbb
R^{\mathtt{Nv}\times\mathtt{Nv}}\) to select which variables to diffuse
and how strongly. The selection matrix often takes the sparse form
\(\underline{P} =
\operatorname{diag}(P_{0}, ..., P_{\mathtt{Nv}-1})\) with mostly zeros.
The reaction term
\(\underline{r}(\underline{u}, \symbfit x) \in \mathbb R^{\mathtt{Nv}}\)
describes the local dynamics of the system and may vary in space, for
instance in parameters or even in the model equations themselves. We
refer to a specific local choice of
\(\underline{r}_{\mathtt{imodel}}(\underline u)\) with fixed parameters as
a model; \(\underline{r}\) refers to the reaction term, i.e., all
models. The source term
\(\underline{s}(t, \symbfit x) \in \mathbb R^{\mathtt{Nv}}\) can be used to
add external influences to the system, for instance to stimulate the
system at specific times and locations.
For no source term, \(\underline s = 0\), homogeneous and isotropic diffusion, \(\symbfit D(\symbfit x) = D = \text{const.}\), and only two variables, \(u = u_0\) and \(v = u_1\), with only \(u\) diffusing, \(P_0 = 1\), \(P_1 = 0\), the system reduces to: