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=xt−1+rxt−1(1−xt−1K)
was simplified to the following form:
x′t=r′x′t−1(1−x′t−1)
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(1−xK)
Here we can apply the following two rescaling rules to both state variable x and time t:
x→ax′
t →βt′
With these replacements, Equation ??? is simplified as
d(ax′)d(βt′)=rax′(1−ax′K)
βα⋅d(ax′)d(βt′)=βα⋅rαx′(1−αx′K)
dx′dt′=rβx′(1−αx′K)
dx′dt′=x′(1−x′)
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.
Simplify the following differential equation by variable rescaling:
dxdt=ax2+bx+c
Simplify the following differential equation by variable rescaling:
dxdt=ax+b
a>0,b>0
Simplify the following two-dimensional differential equation model by variable rescaling:
dxdt=ax(1−x)−bxy
dydt=cy(1−y)−dxy