14.2: Variable Rescaling

Variable rescaling of continuous field models comes with yet another bonus variable, i.e., space, which you can rescale to potentially eliminate more parameters from the model. In a 2-D or higher dimensional space, you can, theoretically, have two or more spatial variables to rescale independently. But the space is usually assumed to be isotropic (i.e., there is no difference among the directions of space) in most spatial models, so it may not be practically meaningful to use different rescaling factors for different spatial variables.

Anyway, here is an example. Let’s try to simplify the following spatial predator-prey model, formulated as a reaction-diffusion system in a two-dimensional space:

$$\begin{align} \dfrac{\partial{r}}{\partial{t}} &=ar-brf+D_{r}\nabla^{2}r \nonumber \\[4pt] &= ar-brf+D_{r} \left(\dfrac{\partial^{2}{r}}{\partial{x^{2}}} +\dfrac{\partial^{2}{r}}{\partial{y^{2}}}\right) \label{(14.20)} \end{align}$$

$$\begin{align} \dfrac{\partial{f}}{\partial{t}} &=-cf+drf +D_{f}\nabla^{2}f \nonumber \\[4pt] &=-cf+drf+D_{f} \left(\dfrac{\partial^{2}f}{\partial{x^{2}}} +\dfrac{\partial^{2}f}{\partial{y^{2}}}\right) \label{(14.21)} \end{align}$$

Here we use $r$ for prey (rabbits) and $f$ for predators (foxes), since $x$ and $y$ are already taken as the spatial coordinates. We can apply the following three rescaling rules to state variables $r$ and $f$, as well as time $t$ and space $x/y$:

$$\begin{align*} r &\rightarrow \alpha{r'} \label{(14.22)} \\[4pt] f &\rightarrow \beta{f'} \label{(14.23)} \\[4pt] t &\rightarrow \gamma{t'} \label{(14.24)} \\[4pt] x, y &\rightarrow \delta x, \delta y \label{(14.25)} \end{align*}$$

With these replacements, the model equations (Equation $\ref{(14.20)}$ and $\ref{(14.21)}$) can be rewritten as follows:

$$\begin{align} \dfrac{\partial(\alpha{r'})}{\partial{\gamma{t'}}} &=a(\alpha{r'}) -b(\alpha{r'})(\beta{f'}) +D_{r} \left(\dfrac{\partial^{2}(\alpha{r'}) }{\partial{(\delta{x'})^{2}}} +\dfrac{\partial^{2}(\alpha{r'})}{\partial{(\delta{y'})^{2}}}\right) \label{(14.26)} \\[4pt] \dfrac{\partial(\beta{f'})}{\partial{(\gamma{t'})}} &=-c(\beta{f'}) +d(\alpha{r'})(\beta{f'}) +D_{f} \left(\dfrac{\partial^{2}(\beta{f'})}{\partial{(\delta{x'})^{2}}} +\dfrac{\partial^{2}(\alpha{r'})}{\partial{(\delta{y'})}}^{2}\right) \label{(14.27)} \end{align}$$

We can collect parameters and rescaling factors together, as follows:

$$\begin{align} \dfrac{\partial{r'}}{\partial{t'}} &= a\gamma{r'} -b{\beta}{\gamma{r' f'}} +\dfrac{D_{r}{\gamma}}{\delta^{2}}(\dfrac{\partial^{2}{r'}}{\partial{x^{'2}}}+\dfrac{\partial^{2}{r'}}{\partial{y^{'2}}})\label{(14.28)} \\[4pt] \dfrac{\partial{f'}}{\partial{t'}} &=-c{\gamma}f'+ d{\alpha{\gamma{r' f'}}} +\dfrac{D_{f}\gamma}{\delta^{2}}(\dfrac{\partial^{2}f'}{\partial{x^{'2}}}+\dfrac{\partial^{2}f'}{\partial{y^{'2}}}) \end{align}$$

Then we can apply, e.g., the following rescaling choices

$$(\alpha, \beta, \gamma, \delta) = \left(\dfrac{a}{d},\dfrac{a}{b}, \dfrac{1}{a}, \sqrt{\dfrac{D_{r}}{a}}\right) \label{(14.30)}$$

to simplify the model equations into

$$\dfrac{\partial{r'}}{\partial{t'}} =r' -r' f' +\nabla^{2}r', \label{(14.31)}$$

$$\dfrac{\partial{f'}}{\partial{t'}} =-e{\gamma{f'}} +r' f' +D_{ratio}\nabla^{2}f' \label{(14.32)}$$

with $e = c/a$ and $D_{ratio} = D_{f}/D_{r}$. The original model had six parameters ($a$, $b$, $c$, $d$, $D_r$, and $D_f$), but we were able to reduce them into just two parameters ($e$ and $D_{ratio}$) with a little additional help from spatial rescaling factor $δ$. This rescaling result also tells us some important information about what matters in this system: It is the ratio between the growth rate of the prey ($a$) and the decay rate of the predators ($c$) (i.e., $e = c/a$), as well as the ratio between their diffusion speeds ($D_{ratio} = D_{f}/D_{r}$), which essentially determines the dynamics of this system. Both of these new parameters make a lot of sense from an ecological viewpoint too.

Exercise $\PageIndex{1}$: Rescaling the Keller-Segel model

Simplify the following Keller-Segel model by variable rescaling:

$$\begin{align*} \dfrac{\partial{a}}{\partial{t}} &= \mu{\nabla}^{2}a -\chi{\nabla} \cdot(a\nabla{c}) \label{(14.33)} \\[4pt] \dfrac{\partial{c}}{\partial{t}} &= D\nabla^{2}c +fa -kc \label{(14.34)} \end{align*}$$