What Is a Logistic Model and What Does Its “Carrying Capacity” Mean?

A logistic model describes growth that begins approximately exponentially but slows as resources become limited. Its carrying capacity is the limiting value that the model approaches over time, not a value it reaches at a finite time.

This is HL-only content from AHL 2.9: Further modelling functions in IB Mathematics: Applications and Interpretation. It may be examined in any HL paper, with a GDC available.

A standard logistic model is

$$f(t)=\frac{L}{1+Ce^{-kt}}, \qquad L,C,k>0.$$

For example, suppose a fish population is modelled by

$$P(t)=\frac{1200}{1+5e^{-0.4t}},$$

where $t$ is measured in years. The carrying capacity is $1200$ fish, meaning the environment is modelled as supporting a long-term population of approximately $1200$.

Initially,

$$P(0)=\frac{1200}{1+5}=200.$$

The model has an S-shaped curve. When the population is well below $L$, the denominator changes rapidly and growth resembles exponential growth. Resource limitations then become increasingly important, reducing the growth rate. At $P=\frac{L}{2}$, the curve changes from concave up to concave down; after this point, the population still increases, but by progressively smaller amounts. This reflects the modelling cycle: growth is constrained by environmental capacity.

After ten years,

$$P(10)=\frac{1200}{1+5e^{-4}}\approx 1.10\times 10^3.$$

The population has increased to approximately $1100$, but its growth slows as it approaches $1200$. The fastest growth occurs at the point of inflexion, when the population is $\frac{L}{2}=600$.

A common misconception is that the model eventually reaches $L$. It does not: $L$ is a horizontal asymptote, so $P(t)<L$ for every finite value of $t$.

In an IB response, identify $L$, state its meaning with contextual units, and distinguish “approaches” from “reaches.” When using a GDC, show the substituted expression before giving the numerical value, retain full precision, and round only the final answer.