AP Calculus BC / Unit 7: Differential Equations
7.9 Logistic Models with Differential Equations
Notes
$$\frac{dP}{dt}=kP\left(1-\frac{P}{L}\right)$$
- $L$ is the carrying capacity; $P\to L$ as $t\to\infty$ (for $P_0>0$).
- Growth is fastest when $\displaystyle P=\frac{L}{2}$ (the inflection point).
Practice
For $\displaystyle \frac{dP}{dt}=0.2P\left(1-\frac{P}{500}\right)$, the carrying capacity is:
It's $L$.
The population grows fastest when $P=$
Half the carrying capacity.
If $P(0)=600$ for that model, then as $t\to\infty$:
Above $L$, $\displaystyle \frac{dP}{dt}<0$.
Ready for more? Take the Unit 7 test.