CALCULUS FOR MODELLING 03
Growth and Decay
Exponential models arise when the instantaneous rate of change is proportional to the current amount.
The main idea
\[\boxed{\frac{dX}{dt}=rX}\]
If \(r\gt0\), positive \(X\) grows. If \(r\lt0\), positive \(X\) decays.
1. Solution
Separating and integrating gives
\[\boxed{X(t)=X_0e^{rt}}\]
2. Why exponential?
\[\frac{d}{dt}e^{rt}=re^{rt}.\]3. Relative growth rate
\[\boxed{\frac1X\frac{dX}{dt}=r}.\]4. Doubling time
For \(r\gt0\),
\[\boxed{T_2=\frac{\ln2}{r}}.\]5. Half-life
For \(dX/dt=-kX\) with \(k\gt0\),
\[\boxed{T_{1/2}=\frac{\ln2}{k}}.\]6. Applications
The model appears in unrestricted population growth, radioactive decay, simple elimination and early-stage epidemic growth.
7. Limitation
Exponential growth cannot usually continue indefinitely because real systems develop constraints.