F. Population Dynamics · Population Dynamics

Limiting Factors: Density-Dependent and Density-Independent

F3.3 F3.2 F2.1 A2.2About 70 minutes

Learning goals

I can explain limiting factors using Liebig's law of the minimum and Shelford's law of tolerance.
I can classify factors as density-dependent or density-independent and justify my choice.
I can explain how density-dependent factors regulate a population near carrying capacity, using per capita birth and death rates.
I can calculate K from per capita birth and death rate equations.

Words to know

limiting factor
Any resource or condition that restricts the growth, abundance or distribution of a population.

law of the minimum
Liebig's principle: growth is limited by the essential resource in shortest supply, not by the total of all resources.

law of tolerance
Shelford's principle: each species survives only within a range of each environmental factor, with an optimum range, zones of stress and limits of tolerance.

density-dependent factor
A factor whose effect on per capita births or deaths grows stronger as population density increases (competition, predation, disease, parasites, waste build-up).

density-independent factor
A factor that affects a population regardless of its density (extreme weather, fire, flood, drought).

population regulation
The tendency of density-dependent factors to push a population back toward K, a form of negative feedback.

eutrophication
Over-enrichment of a lake with nutrients (especially phosphorus), causing algal blooms and oxygen depletion.

What stops a population?

The logistic model says populations level off at K. But what actually holds them there? This lesson names the limiting factors and sorts them into two families that behave very differently.

A limiting factor is any resource or condition that restricts a population's growth or distribution: food, water, light, nutrients, nesting sites, temperature, predators, disease.

Two classic laws

Liebig's law of the minimum. In the 1800s the German chemist Justus von Liebig noticed that a crop's growth is limited by whichever essential nutrient is scarcest. Adding extra nitrogen does nothing if phosphorus is what is missing. Think of a barrel whose staves are different heights: it can only fill to the level of the shortest stave.

Shelford's law of tolerance. In 1911 the American ecologist Victor Shelford added that too much can limit a population as well as too little. For every factor (temperature, pH, salinity, moisture), a species has an optimum range where it thrives, zones of physiological stress where it survives but reproduces poorly, and limits of tolerance beyond which it cannot survive. Brook trout, for example, need cold, well-oxygenated water; warm summer streams can exceed their tolerance even when food is plentiful.

Shelford's law of tolerance. A population is largest in the optimum range, smaller in the zones of stress, and absent beyond its limits of tolerance.

Worked example

Let's read a Canadian whole-lake experiment

At the Experimental Lakes Area in northwestern Ontario, a team led by David Schindler divided Lake 226 into two basins with a plastic curtain in 1973. One basin received carbon and nitrogen; the other received carbon, nitrogen and phosphorus. Only the phosphorus basin developed a large algal bloom. What was the limiting factor, and which law does this illustrate?

Identify the variable: both basins received carbon and nitrogen; only one received phosphorus.
Observe the result: only the basin with phosphorus bloomed.
Conclude: algae in the lake were limited by phosphorus, the nutrient in shortest supply. Adding the others did not help.
Law: this is Liebig's law of the minimum in a whole ecosystem.
Impact (A2.2): results like these helped persuade governments in Canada and elsewhere to limit phosphates in laundry detergents and to treat sewage to remove phosphorus, reducing eutrophication in lakes such as Lake Erie.

Phosphorus was the limiting factor, an example of Liebig's law of the minimum.

Density-dependent factors

A factor is density-dependent when its effect on per capita births or deaths becomes stronger as the population becomes denser. Examples:

Intraspecific competition for food, water, nest sites or light. Crowded white-tailed deer browse their winter food supply down, and fawns born the next spring are smaller.
Predation: many predators catch a larger share of their prey when prey are abundant, or switch to the most common prey.
Disease and parasites spread faster when hosts are crowded. Winter ticks, for instance, can build up when moose are dense, weakening the animals.
Territoriality and stress: animals without territories may not breed.
Accumulation of wastes: yeast in a closed culture are poisoned by their own ethanol.

Because their effect rises with density, density-dependent factors act as negative feedback. They regulate a population, pushing it back toward K. In the logistic model, they are what produce the (K − N) ÷ K brake.

Density-dependent regulation. As N rises, per capita births fall and per capita deaths rise. They cross at the carrying capacity: here b = 0.60 − 0.0010N and d = 0.10 + 0.0010N meet at N = 250.

Worked example

Let's find K from birth and death rates

In a population of grey squirrels, the per capita birth rate is b = 0.60 − 0.0010N and the per capita death rate is d = 0.10 + 0.0010N (both per year). (a) Find K. (b) Find the per capita growth rate when N = 100 and when N = 400.

(a) At K, births equal deaths: 0.60 − 0.0010N = 0.10 + 0.0010N.
Rearrange: 0.50 = 0.0020N, so N = 250 = K.
(b) N = 100: b = 0.60 − 0.10 = 0.50; d = 0.10 + 0.10 = 0.20. Per capita growth = b − d = +0.30 per year (grows).
N = 400: b = 0.60 − 0.40 = 0.20; d = 0.10 + 0.40 = 0.50. Per capita growth = −0.30 per year (shrinks).
Either way, the population is pushed toward 250: that is regulation.

(a) K = 250. (b) +0.30 per year at N = 100; −0.30 per year at N = 400.

Density-independent factors

A factor is density-independent when it kills (or prevents births in) about the same proportion of a population whatever its density. These are usually abiotic events:

Extreme weather: a late spring frost that kills apple blossoms in an orchard; a cold snap; an ice storm, such as the January 1998 ice storm that damaged huge numbers of trees across eastern Ontario and southern Quebec.
Fire, flood and drought.
Many human impacts: an oil spill or the draining of a wetland.

A Canadian example: mountain pine beetle larvae spend the winter under the bark of pine trees. Long spells of extreme cold (near −40 °C), especially early in winter before the larvae build up their natural antifreeze, can kill most of them, whether the beetle population is large or small. A run of milder winters is one reason outbreaks spread so far in British Columbia and into Alberta.

Density-independent factors do not regulate a population toward K. They cause sudden crashes or booms, which is why many insect and small-mammal populations fluctuate so much.

	Density-dependent	Density-independent
Effect on per capita rates	Stronger at high density	Same at any density
Usually	Biotic (competition, predators, disease)	Abiotic (weather, fire, flood)
Regulates toward K?	Yes (negative feedback)	No: causes sudden changes
Examples	Food shortage in crowded deer; disease in dense colonies	Ice storm; extreme cold killing pine beetle larvae

Comparing the two families of limiting factors.

The test question to ask yourself

"If the population were twice as dense, would this factor kill (or prevent the birth of) a larger fraction?" If yes, it is density-dependent. If the fraction stays about the same, it is density-independent. Some factors are mixed: in a flood, a crowded population may be forced into poorer, more exposed sites, so more die.

Just for fun

Liebig's barrel in real life: you can have all the flour, eggs and sugar in the world, but if you have no baking powder, the cake's growth is limited by the minimum.

Hands-on activity

Liebig's law with seeds

You need: about 30 dried beans or lentils from the kitchen, 4 small containers (yogurt cups with holes poked in the bottom by an adult), potting soil and sand, a ruler, water, a sunny window, labels. Ask an adult to poke the drainage holes.

Soak the seeds in water overnight.
Set up four cups with 6 seeds each: A potting soil, watered regularly, on the sunny windowsill; B potting soil, watered regularly, in a dark cupboard; C sand only (few nutrients), watered regularly, in the window; D potting soil in the window, but watered only once.
After 10 to 14 days, count the seedlings in each cup and measure their heights.
For each cup, identify the limiting factor and explain your results with Liebig's law.
Is any of these factors density-dependent? Plant 3 seeds in one extra cup and 15 in another (same soil, light and water) to find out.

What to look for: A should grow best. B's seedlings may grow tall and pale at first, using food stored in the seed, but then stop: light limits them. C grows at first (seed reserves) but stays small and yellowish: nutrients. D wilts: water. In the crowding test, seedlings in the 15-seed cup are usually thinner and shorter on average: intraspecific competition is density-dependent. Wash hands after handling soil.

Your turn

Classify each as density-dependent or density-independent: (a) a forest fire, (b) a viral disease in a crowded bat colony, (c) competition for nest cavities, (d) a summer drought, (e) an owl population that eats more voles when voles are common.

State Liebig's law of the minimum in your own words.

How does Shelford's law add to Liebig's?

Per capita births b = 0.50 − 0.002N and deaths d = 0.10 + 0.002N. Find K.

Why do density-dependent factors regulate a population, while density-independent factors do not?

Explain why extreme winter cold is considered density-independent for mountain pine beetles.

More practice: warm-up

What is a limiting factor?

Are density-dependent factors usually biotic or abiotic?

Give one density-independent factor.

What nutrient limited algae in the Lake 226 experiment?

In the law of tolerance, where is a population largest?

More practice: on your own

A deer population of 800 loses 10% of its fawns in a hard winter; a population of 1600 in an identical area loses 25%. Is the winter's effect density-dependent? Explain.

Using b = 0.80 − 0.004N and d = 0.20 + 0.002N, find K and the net per capita rate at N = 50.

Lake trout need water below about 15 °C with plenty of oxygen. Use the law of tolerance to explain why a warming lake might lose its lake trout even if food is plentiful.

Explain why many insect populations fluctuate wildly from year to year, while populations of large mammals are often steadier.

Give an example of a factor that can act in both a density-dependent and density-independent way, and explain.

Challenge

A population has per capita birth rate b = 0.9 − 0.003N and death rate d = 0.1 + 0.001N. (a) Find K. (b) Show that the per capita growth rate r = b − d can be written in the logistic form rₘₐₓ(1 − N/K) and give rₘₐₓ. (c) What is the maximum dN/dt?

Design an experiment to test whether disease in a laboratory population of fruit flies is density-dependent. Include the independent and dependent variables, controls and the result that would support density dependence.

Before the Lake 226 experiment, some detergent makers argued that carbon, not phosphorus, limited algae. Explain why a whole-lake experiment was more persuasive than small bottle experiments, and identify one limitation of the whole-lake design.

Check your understanding

Quiz

Which factor is density-dependent?

Liebig's law of the minimum states that growth is limited by:

If b = 0.70 − 0.002N and d = 0.30 + 0.002N, what is K?

Density-independent factors:

The Lake 226 experiment at the Experimental Lakes Area showed that algal growth was limited by:

According to Shelford's law of tolerance, a population outside its limits of tolerance will:

For parents and teachers

Big idea (F3.3): limiting factors stop populations growing. Density-dependent factors (competition, predation, disease) grow stronger with crowding and regulate populations near K; density-independent factors (weather, fire, flood) act regardless of density and cause sudden changes. Liebig's and Shelford's laws describe how resources and conditions limit growth.

How to help: the seed activity takes two weeks of occasional checking; put a reminder on the fridge. Ask your teen the key test question: "If the population were twice as dense, would this kill a larger fraction?"

Common difficulties:

Assuming anything that kills many individuals is density-dependent. It is the fraction that matters.
Algebra when finding K from birth and death equations.

Sources: Schindler (1974), Science, on Lake 226 at the Experimental Lakes Area; Shelford (1911); Liebig's law from standard ecology texts; mountain pine beetle cold-mortality information from Natural Resources Canada.