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Mathematics LibreTexts

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

xt=xt1+rxt1(1xt1K)

was simplified to the following form:

xt=rxt1(1xt1)

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:

dxdt=rx(1xK)

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

xax

t βt

With these replacements, Equation ??? is simplified as

d(ax)d(βt)=rax(1axK)

βαd(ax)d(βt)=βαrαx(1αxK)

dxdt=rβx(1αxK)

dxdt=x(1x)

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 7.3.1

Simplify the following differential equation by variable rescaling:

dxdt=ax2+bx+c

Exercise 7.3.2

Simplify the following differential equation by variable rescaling:

dxdt=ax+b

a>0,b>0

Exercise 7.3.3

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

dxdt=ax(1x)bxy

dydt=cy(1y)dxy


This page titled 7.3: Variable Rescaling of Continuous-Time Models is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Hiroki Sayama (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.

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