$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.3: Variable Rescaling of Continuous-Time Models

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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:

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

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