Exponential growth: dN/dt = rN. Logistic growth: dN/dt = rN(K − N)/K, which slows growth as N nears K. Calculate rates at a given N and read the S-curve.
In Exponential and Logistic Growth you saw the J-curve and the S-curve. Now we use their equations to calculate how fast a population is growing at a given size.
🔑 Key idea
Change in population: $\dfrac{dN}{dt} = B - D$ (births minus deaths per unit time). Exponential growth: $$\frac{dN}{dt} = rN$$ Logistic growth: $$\frac{dN}{dt} = rN\,\frac{K - N}{K}$$ $r$ is the per-capita growth rate and $K$ the carrying capacity. Logistic growth slows as $N$ nears $K$ and is fastest at $N = K/2$.
The S-curve. The dashed line marks K/2, where growth is fastest. Change r and K.
💡 Why it works
The factor $(K - N)/K$ is the share of the carrying capacity still unused. When $N$ is small, it is close to 1, so growth is almost exponential. When $N = K$, it is 0, so growth stops: births balance deaths. In between, growth is the product of two things pulling in opposite directions: more individuals to have offspring ($N$) and less room left ($K - N$). Their product $N(K - N)$ is largest halfway, at $N = K/2$.
Logistic growth rate against N: an upside-down U with its top at K/2.
Worked examples
✏️ Worked example: Exponential growth
A population of 400 rabbits has r = 0.3 per year with no limits. How fast is it growing?
Answer: 120 rabbits per year (and faster as it grows).
✏️ Worked example: Logistic growth
The same rabbits live where K = 1,000. Find the growth rate at N = 400 and at N = 900.
N = 400: $0.3 \times 400 \times \frac{1000 - 400}{1000} = 120 \times 0.6 = 72$ per year.
N = 900: $0.3 \times 900 \times \frac{100}{1000} = 270 \times 0.1 = 27$ per year.
Fastest at N = 500: $0.3 \times 500 \times 0.5 = 75$ per year.
Answer: 72 rabbits per year at N = 400 and 27 at N = 900.
Where does $r$ come from? It is the per-capita rate: births minus deaths, divided by the population size. A herd of 1,000 deer with 300 births and 100 deaths in a year has $r = (300 - 100)/1{,}000 = 0.2$ per year, and it grows by $dN/dt = 200$ deer per year. In the logistic model, what slows growth near $K$ are density-dependent factors you met before: competition for food and space, disease and predators all bite harder in a crowded population, raising deaths or lowering births until they balance.
⚠️ Common mistake
Plugging in N = K and expecting the fastest growth. At $N = K$ the factor $(K - N)/K$ is zero, so growth is zero. Growth is fastest at $N = K/2$. Check: the logistic rate can never exceed the exponential rate $rN$ at the same $N$. And if $N$ is above $K$, for example after a good year, the formula gives a negative rate: the population shrinks back toward $K$.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 3 XP.