Carrying Capacity, Limiting Factors and Growth Curves

Carrying capacity and limiting factors

A population cannot grow for ever. Every habitat has a limited supply of food, water, space and shelter, so there is a maximum number of individuals it can support year after year.

Anything that holds a population back is a limiting factor. Limiting factors are sorted into two groups, depending on whether their effect changes with how crowded the population is.

Abiotic factors such as water, oxygen, space, temperature and soil can also limit a population, for example by setting how much food a habitat can grow. Take red deer in Killarney National Park. In a crowded herd, there is less grazing per animal and disease spreads more easily, so food and disease are density-dependent. A harsh winter storm kills deer whether the herd is large or small, so it is density-independent.

J-curves and S-curves

If a few individuals arrive in a habitat with plenty of resources, they breed freely and the population grows by a constant percentage each time period. This is exponential growth, and it gives a J-shaped curve.

When resources begin to run out, growth slows and the population levels off around K. This gives an S-shaped (sigmoid) curve.

The S-curve has three phases:

1. Lag phase: few individuals, slow start while the population adjusts to its new habitat.

2. Exponential phase: rapid increase, resources plentiful.

3. Stationary phase: births equal deaths, and the population fluctuates around K.

A pure J-curve cannot last. It is typical of a short burst, such as bacteria in fresh broth or an invasive species arriving in a new habitat. Over a long period, natural populations show S-shaped growth, because limiting factors always catch up.

Working with population data

Exam questions give you a table or graph and ask for a handful of standard things.

• Carrying capacity: the value at which the population levels off, so read the plateau.

• Average rate of increase: $\frac{\text{change in population}}{\text{change in time}}$.

• Phases: find where the slope changes.

A farmer's field shows the same idea. Total gain is the number of sheep multiplied by the gain per sheep. With 10 sheep: $10 \times 8 = 80$ kg. With 30: $30 \times 5 = 150$ kg. With 60: $60 \times 1 = 60$ kg. Total gain rises, then falls: beyond a certain stocking level, competition for grass reduces what each animal gets. At 60 sheep each animal gains only 1 kg, and at 80 some lose weight, so the field is at or beyond its carrying capacity. The best stocking level is near the peak (around 30 sheep), where the total is highest and each animal still does well. Carrying capacity can be raised by reseeding, adding fertiliser, or providing supplementary feed.

The logistic equation

The S-curve can be described mathematically by the logistic equation:

• $N$ is the population size, $K$ the carrying capacity.

• $r$ is the intrinsic rate of increase, the growth rate per individual when resources are unlimited.

• $\frac{K-N}{K}$ is the fraction of K still available. It is close to 1 when $N$ is small (almost no resistance) and close to 0 when $N$ nears K. When $N = K$, the bracket is 0, so $\frac{dN}{dt} = 0$ and growth stops.

Growth is fastest at $N = K/2$ (here 2,500), where the product $N(K-N)$ is largest: there are plenty of breeders and plenty of room. This is the idea behind a sustainable yield: harvest no more than the population's growth per year, ideally near $K/2$, so the stock replaces itself. Harvesting at or near K gives little surplus, and overfishing far below $K/2$ means few breeders to replace what is taken.

Predators, competitors and moving targets

Real populations rarely sit neatly at a fixed K. Carrying capacity changes with the seasons, the food supply and other species.

Predator–prey cycles. This is a classic textbook example (an Irish equivalent would be foxes and rabbits on grassland or hedgerow). In a boreal forest, snowshoe hare numbers rise when food is plentiful. Lynx, which eat hares, then increase about two years later, because they need time to find food and reproduce. More lynx reduce hares, and then the lynx starve and fall in turn. Predation is density-dependent: the more hares there are, the easier they are to catch, and the more predators the food supports. The result is an oscillating curve, not a smooth S-curve.

Hence K for the hares is not fixed: it depends on vegetation quality, which varies with weather, and on how many lynx are present.

Competition and disease. Red and grey squirrels compete for the same food, but greys use it more efficiently and can feed on unripe seeds, so greys win. Greys also carry the squirrelpox virus, to which they are immune, but which is usually fatal to reds. A disease therefore acts differently on the two species: for the greys it is almost harmless, for the reds it is a major, density-dependent mortality factor. The effective carrying capacity for reds falls, and the population declines.

Management measures include:

• Grey squirrel control (trapping or culling in red squirrel areas), which removes the competitor and the source of virus.

• Habitat protection and planting conifers or mixed woodland where reds do better than greys, which raises the carrying capacity for reds.

Back to the start: grey squirrels can use food better and carry a virus that kills reds, which lowers the carrying capacity for red squirrels where the two species share a wood.

Summary

• Carrying capacity (K) is the maximum population an environment can support indefinitely.

• Density-dependent factors (food, competition, predation, disease) act more strongly as density rises. Density-independent factors (frost, flood, fire) do not depend on density.

• Unlimited growth gives a J-curve. Limited resources give an S-curve with lag, exponential and stationary phases, and this is the usual pattern over a long period.

• Read K from the plateau, and find a rate using change in population ÷ change in time.

• The logistic equation $\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right)$ gives zero growth at $N = K$ and fastest growth at $K/2$. This is the basis of sustainable yield.

• Carrying capacity is not fixed. It shifts with food supply, predators, competitors and disease.