Modelling Rates of Change
Modelling Rates of Change
In mathematical modelling, verbal increase and decrease statements are translated into a signed differential equation. Define every variable and its units before forming the balance.
\(\begin{aligned}\dfrac{\mathrm{d}x}{\mathrm{d}t}&=\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)_{\!\mathrm{in}}-\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)_{\!\mathrm{out}}\\ &=r_{\mathrm{in}}-r_{\mathrm{out}}\end{aligned}\)
The subscripts “in” and “out” label the non-negative rates at which the quantity is added and removed: \(r_{\mathrm{in}}=\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)_{\!\mathrm{in}}\) and \(r_{\mathrm{out}}=\left(\dfrac{\mathrm{d}x}{\mathrm{d}t}\right)_{\!\mathrm{out}}\). The unlabelled \(\dfrac{\mathrm{d}x}{\mathrm{d}t}\) is the signed net rate of change of \(x\). Subtract the removal rate once; do not first make it negative.
Typical proportional models
| Verbal model | Differential equation | Conditions |
|---|---|---|
| The signed rate of change of \(N\) is proportional to \(N-q\). | \(\displaystyle\frac{\mathrm{d}N}{\mathrm{d}t}=k(N-q)\) | \(k>0\); sign depends on \(N-q\). |
| The rate-of-decrease magnitude of \(y\) is inversely proportional to \(y(100-y)\). | \(\displaystyle-\frac{\mathrm{d}y}{\mathrm{d}t}=\frac{k}{y(100-y)}\) | \(k>0,\ 0<y<100\). |
| For remaining proportion \(x\), the decrease-rate magnitude is proportional to \(x(1-x)\) | \(\displaystyle-\frac{\mathrm{d}x}{\mathrm{d}t}=kx(1-x)\) | \(k>0,\ 0\leq x\leq1\). |
If the quantity is decreasing, its signed derivative must be negative. Every term in the same balance must have units of quantity per unit time.
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