The logistic model represents population growth when the per-capita growth rate declines as population size increases.

Let $K$ be the carrying capacity, the population size at which the model predicts zero net growth under a particular set of environmental conditions. The logistic equation is

$$\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).$$

The factor

$$1-\frac{N}{K}$$

reduces the effective per-capita growth rate as $N$ approaches $K$.

• If $N\ll K$, the factor is near 1 and growth is approximately exponential.
• If $N=K$, net growth is zero.
• If $N>K$, the model predicts negative growth until population size declines toward $K$.

Worked example

Suppose $r=0.5\ \text{yr}^{-1}$, $K=1000$ and $N=200$. Then

$$\frac{dN}{dt}=0.5(200)\left(1-\frac{200}{1000}\right) =100(0.8)=80\ \text{individuals/yr}.$$

At $N=900$,

$$\frac{dN}{dt}=0.5(900)(0.1)=45\ \text{individuals/yr},$$

so absolute growth has slowed despite the larger population.

Carrying capacity is not a permanent property of a species. It depends on the environment: food, water, habitat, competitors, climate and other conditions can change $K$.

The logistic model is therefore a useful minimal model of density limitation, not a claim that real populations always follow a smooth S-shaped curve or remain exactly at one fixed carrying capacity.