CALCULUS FOR MODELLING 04
Logistic Growth
The logistic model reduces growth as a population approaches a limiting capacity.
The model
\[\boxed{\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)}\]
Here \(r\gt0\) is the intrinsic growth rate and \(K\gt0\) is the carrying capacity.
1. Why modify exponential growth?
The factor \(1-P/K\) reduces growth as \(P\) increases.
2. Small populations
If \(P\ll K\), then \(P'\approx rP\), so early logistic growth resembles exponential growth.
3. Near carrying capacity
When \(P\approx K\), growth slows toward zero.
4. Equilibria
\[\boxed{P^*=0\quad\text{or}\quad P^*=K}\]
5. Stability
For \(0\lt P\lt K\), \(P'\gt0\). For \(P\gt K\), \(P'\lt0\). Thus positive solutions tend toward \(K\).
6. Maximum growth rate
\[\boxed{P=\frac K2}.\]7. Explicit solution
For \(P_0\gt0\),
\[\boxed{P(t)=\frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}}\]
8. Shape and interpretation
The solution is typically S-shaped below \(K\): near-exponential growth, fastest growth near \(K/2\), then flattening toward \(K\).
9. Limitation
Real populations may have changing resources, delays, seasonal effects or additional structure.