Logistic models: carrying capacity and fastest growth

Understand the idea

A logistic model describes growth whose rate depends on both the current quantity and the remaining capacity. For a positive intrinsic growth parameter, a population between zero and its carrying capacity increases, while a population above capacity decreases toward it. The equilibria and the sign of the derivative explain the long-term behavior.

In $\displaystyle P'=rP(1-P/K)$, the positive constant $\displaystyle K$ is the carrying capacity. For $\displaystyle 0<P<K$, the rate is a downward-opening quadratic in $\displaystyle P$ and is largest at $\displaystyle P=K/2$. For an increasing non-equilibrium solution, this is also the inflection level of the population graph.

An equivalent form is $\displaystyle P'=kP(K-P)$, where $\displaystyle r=kK$. Distinguish these parameters instead of copying the coefficient into a memorized solution. Initial values $\displaystyle P=0$ and $\displaystyle P=K$ give constant solutions, which can be lost if separation divides by $\displaystyle P(K-P)$.

Worked example

A population satisfies $\displaystyle P'=0.3P(1-P/600)$ and $\displaystyle P(0)=100$. Find its carrying capacity, maximum growth rate, and the time it reaches half capacity.

1. The carrying capacity is $\displaystyle K=600$. The greatest growth occurs at $\displaystyle P=300$.
2. The maximum rate is $\displaystyle 0.3(300)(1-300/600)=45$ population units per time unit.
3. The initial condition gives $\displaystyle A=(600-100)/100=5$, so $\displaystyle P(t)=600/(1+5e^{-0.3t})$.
4. Setting $\displaystyle P=300$ gives $\displaystyle 5e^{-0.3t}=1$, hence $\displaystyle t=\ln(5)/0.3$.