7.3: Variable Rescaling of Continuous-Time Models

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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:

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

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