In Limiting Factors and Carrying Capacity you saw that populations cannot grow forever. Two simple models describe how they grow, and they use the exponential growth you know from algebra.
Key idea
With unlimited resources, a population grows exponentially: $\frac{dN}{dt} = rN$, a J-shaped curve. As resources run short, growth slows and the population levels off at the carrying capacity K: logistic growth, $\frac{dN}{dt} = rN\frac{K - N}{K}$, an S-shaped curve.
Why it works
In exponential growth, every individual adds offspring at the same rate r, so more individuals means faster growth: the rate is proportional to N. In logistic growth the extra factor $\frac{K - N}{K}$ is the share of the habitat still unused. When N is small it is nearly 1, so growth looks exponential; when N reaches K it is 0, so growth stops.
Worked example: Exponential growth rate
A population of 400 rabbits has r = 0.3 per year with plenty of food. How fast is it growing?
- $\frac{dN}{dt} = rN = 0.3 \times 400$.
Answer: 120 rabbits per year.
Worked example: Logistic growth rate
The same population lives where K = 1,000. How fast does it grow at N = 400? At N = 900?
- At 400: $0.3 \times 400 \times \frac{1000 - 400}{1000} = 120 \times 0.6 = 72$ per year.
- At 900: $0.3 \times 900 \times \frac{100}{1000} = 27$ per year.
Answer: 72 per year at N = 400, and only 27 per year at N = 900.
Why it works
Why is K/2 the fastest point? Growth is r × N × (share of room left). At N = K/2 there are many individuals and half the room is still free. Below that there are too few individuals; above it there is too little room. The product is largest exactly in the middle.
Common mistake
It is tempting to think growth is fastest when the population is biggest, near K. In logistic growth it is fastest at about half of K, then slows as the population fills its habitat. Check with numbers: at N = K the growth rate is $rK \times 0 = 0$.
Real populations only roughly follow these models, and many overshoot K and dip back. See how predators and prey push each other up and down in the Predators and Prey simulation.