7.3: Variable Rescaling of Continuous-Time Models

Variable rescaling of continuous-time models has one distinct difference from variable rescaling that of discrete-time models. That is, you get one more variable you can rescale: time. This may allow you to eliminate one more parameters from your model compared to discrete-time cases.

Here is an example: the logistic growth model. Remember that its discrete-time version

$$x_{t}=x_{t-1} +rx_{t-1}(1-\frac{x_{t-1}}{K})\label{(7.26)}$$

was simplified to the following form:

$$x'_{t} =r'x'_{t-1}(1-x'_{t-1}) \label{(7.27)}$$

There was still one parameter $(r')$ remaining in the model even after rescaling. In contrast, consider a continuous-time version of the same logistic growth model:

$$\frac{dx}{dt} =rx(1-\frac{x}{K}) \label{(7.28)}$$

Here we can apply the following two rescaling rules to both state variable $x$ and time $t$:

$$x \rightarrow ax' \label{(7.29}$$

$$t\ \rightarrow \beta {t'}\label{(7.30)}$$

With these replacements, Equation $\ref{(7.28)}$ is simplified as

$$\frac{d(ax')}{d(\beta {t'})} =rax'(1-\frac{ax'}{K}) \label{(7.31)}$$

$$\frac{\beta}{\alpha} \cdot \frac{d(ax')}{d(\beta{t'})} = \frac{\beta}{\alpha} \cdot r\alpha{x'} (1-\frac{\alpha{x'}}{K}) \label{(7.32)}$$

$$\frac{dx'}{dt'} = r\beta{x'}(1-\frac{\alpha{x'}}{K}) \label{(7.33)}$$

$$\frac{dx'}{dt'} =x'(1-x')\label{(7.34)}$$

with $α = K$ and $β = 1/r$. Note that the final result doesn’t contain any parameter left! This means that, unlike its discrete-time counterpart, a continuous-time logistic growth model doesn’t change its essential behavior when the model parameters $(r, K)$ are varied. They only change the scaling of trajectories along the $t$ or $x$ axis.

Exercise $\PageIndex{1}$

Simplify the following differential equation by variable rescaling:

$$\frac{dx}{dt} =ax^{2} +bx+c \label{(7.35)}$$

Exercise $\PageIndex{2}$

Simplify the following differential equation by variable rescaling:

$$\frac{dx}{dt}=\frac{a}{x+b} \label{(7.36)}$$

$$a >0, b >0 \label{(7.37)}$$

Exercise $\PageIndex{3}$

Simplify the following two-dimensional differential equation model by variable rescaling:

$$\frac{dx}{dt} =ax(1-x)-bxy \label{(7.38)}$$

$$\frac{dy}{dt} =cy(1-y)-dxy \label{(7.39)}$$