1Carrying capacity

The carrying capacity ($K$) is the number of individuals of a species that an ecosystem's total available resources can sustain. "Sustain" matters: a population can briefly rise above $K$, but it then uses resources faster than they are replaced, and numbers fall back.

$K$ is not a fixed property of the species. It belongs to the species in a particular place at a particular time. A wet year that doubles the grass on a plain raises $K$ for the antelope there; a drought lowers it. Two woods of the same size can have very different $K$ for owls if one has more old trees with nesting holes.

Population density is the number of individuals per unit area or volume. Density and resources work on each other in both directions: available resources set how dense a population can become, and a denser population leaves fewer resources for each individual.

2Limiting factors

Anything that slows growth by raising the death rate or lowering the birth rate is a limiting factor. There are two kinds, and the test between them is whether the effect gets stronger as density rises.

Density-dependent factors have a bigger effect when the population is dense:

• Competition for food, water, light, nesting sites or mates: more individuals, smaller shares.
• Predation: predators often concentrate on abundant prey.
• Disease and parasites: they spread faster when hosts are close together.
• Accumulation of wastes: yeast in a sealed flask is poisoned by its own ethanol.

Density-independent factors affect the population whatever its density:

• weather extremes (a hard frost, a heatwave), floods, droughts, fires, volcanic eruptions.

A frost that kills 40% of a beetle population kills about 40% whether there are 100 beetles or 10,000. A disease that kills 5% of a sparse population and 45% of a crowded one is density-dependent.

Worked example: tell them apart from data (invented). Two factors were measured in plots of field mice at different densities.

Parasite deaths rise with density: density-dependent. Flood deaths stay near 30% at every density: density-independent.

Density-dependent factors bite harder as a population gets denser; density-independent factors take the same share whatever the density.

3The logistic equation

The formula sheet gives

$$\frac{dN}{dt} = r_{max}N\left(\frac{K-N}{K}\right)$$

where $N$ is population size, $r_{max}$ the maximum per capita growth rate, and $K$ the carrying capacity. The first part, $r_{max}N$, is exponential growth (8.3). The bracket is the fraction of the carrying capacity still unused, and it shrinks as $N$ grows.

Worked example (invented). A deer population has $r_{max} = 0.3$ per year in a forest with $K = 1{,}200$.

Read the table as a story. A small herd grows slowly because there are few deer to reproduce. Growth is fastest at $N = 600$, which is $\frac{K}{2}$: enough deer to breed, still plenty of food. At 1,000 the shortage of food holds growth back. At $K$ growth stops. Above $K$ the rate is negative and the population shrinks back. Keep the unrounded fraction in the calculation; rounding 0.83 early gives 49.8 instead of 50.

Figure 1 shows the same deer two ways. Panel (a) is the familiar S-shaped (sigmoid) curve: slow at first, steepest at $\frac{K}{2}$, flattening at $K$. Panel (b) plots the growth rate itself against $N$, and the peak at 600 is the steepest point of panel (a).

Percent change (skill 5.A). If the deer rise from 600 to 690 in a year, the percent change is $\frac{690-600}{600} \times 100 = 15\%$. Percent change is always (new minus old) divided by old.

4Real populations near K

Real populations rarely sit exactly on the curve. Many overshoot $K$, crash below it, and oscillate around it, especially when there is a delay between the population rising and the resource running short (a summer's births arrive before the winter's food shortage). $K$ itself moves when the environment changes. On a graph, read $K$ as the level around which the population settles, not as the single highest point it ever reached.