Population Ecology and Density Effects
In 1944, 29 reindeer were released on St. Matthew Island, Alaska, a remote island with plenty of lichen to eat. Within 20 years there were about 6,000. Then the herd crashed to a few dozen animals. Why did a population that grew so fast fall so hard? By the end of this page you will be able to explain it.
Exponential growth
A population grows when more individuals are born than die. Under ideal conditions (unlimited food, space, and no predators), every individual can reproduce at its maximum rate. More individuals means more births, which means even more individuals. The population grows faster and faster.
This is exponential growth. The graph is a J-shaped curve.
The growth rate is the change in population size per unit time, $\dfrac{dN}{dt}$. For exponential growth:
Why does the rate depend on $N$? Each individual adds offspring at the same rate, so more individuals add more offspring in total.
For exponential growth, $\dfrac{dN}{dt} = r_{max}N$. A bacterial population has $N = 100$ cells and $r_{max} = 0.05$ per day. Calculate $\dfrac{dN}{dt}$ and state its…
For exponential growth, $\dfrac{dN}{dt} = r_{max}N$. A bacterial population has $N = 100$ cells and $r_{max} = 0.05$ per day. Calculate $\dfrac{dN}{dt}$ and state its units.
Logistic growth and carrying capacity
Exponential growth cannot last. Food, water, nesting sites, and space run out. The largest population an environment can support over time is its carrying capacity, written $K$.
As $N$ gets close to $K$, the growth rate drops. The curve is S-shaped (sigmoid) and levels off near $K$. This is logistic growth.
The logistic equation adds a braking term:
Let's see how the growth rate changes with $N$. Take $r_{max} = 0.4$ per year and $K = 1{,}000$.
| $N$ | $\dfrac{K-N}{K}$ | $\dfrac{dN}{dt}$ (individuals/year) |
|---|---|---|
| 100 | 0.9 | 36 |
| 500 | 0.5 | 100 |
| 900 | 0.1 | 36 |
| 1,000 | 0 | 0 |
The growth rate is greatest at $N = K/2$, where the population is half of carrying capacity. Below that, there are few individuals reproducing. Above that, resources are increasingly scarce.
Define carrying capacity*. Then describe the shape of a graph of exponential growth and of logistic growth, and state which one levels off.
Define carrying capacity. Then describe the shape of a graph of exponential growth and of logistic growth, and state which one levels off.
A population of deer introduced to an island was counted every two years. | Time (years) | Population size | |---|---| | 0 | 50 | | 2 | 100 | | 4 | 250 | | 6 | 500 | | 8…
A population of deer introduced to an island was counted every two years.
| Time (years) | Population size |
|---|---|
| 0 | 50 |
| 2 | 100 |
| 4 | 250 |
| 6 | 500 |
| 8 | 800 |
| 10 | 950 |
| 12 | 1,000 |
(a) Identify the growth model and estimate the carrying capacity. (b) Estimate when the population is growing fastest, and explain why. (c) Explain why growth slows as the population approaches 1,000.
The logistic growth equation is $\dfrac{dN}{dt} = r_{max}N\left(\dfrac{K-N}{K}\right)$. A population has $r_{max} = 0.4$ per year and $K = 1{,}000$. Calculate…
The logistic growth equation is $\dfrac{dN}{dt} = r_{max}N\left(\dfrac{K-N}{K}\right)$. A population has $r_{max} = 0.4$ per year and $K = 1{,}000$. Calculate $\dfrac{dN}{dt}$ when $N = 200$, $N = 500$, and $N = 1{,}000$. Explain what the pattern tells you about where the population grows fastest.
Density-dependent and density-independent factors
What actually pushes a population toward $K$? Limiting factors. They fall into two groups.
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Density-dependent factors have a stronger effect as the population becomes more crowded. Examples: competition for food, disease spreading in crowded herds, predation when prey are abundant, buildup of waste.
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Density-independent factors affect the population by the same proportion no matter how crowded it is. Examples: hurricanes, wildfires, floods, freezes, volcanic eruptions.
Density-dependent factors act like a thermostat: they are a form of negative feedback. Higher density leads to lower birth rates or higher death rates, which brings density back down.
Where the lines cross, births equal deaths and the population holds steady at $K$.
Population data can show which kind of factor is at work. A population held steady near one value, rising when below it and falling when above it, suggests density-dependent control. A population that swings widely with the weather suggests density-independent control.
Classify each limiting factor as density-dependent or density-independent: (a) competition for food; (b) a hurricane that destroys a coastal bird colony; (c) a disease…
Classify each limiting factor as density-dependent or density-independent:
(a) competition for food;
(b) a hurricane that destroys a coastal bird colony;
(c) a disease that spreads faster in crowded herds;
(d) a wildfire in a dry forest;
(e) predation that increases when prey are abundant.
Flour beetles were raised at different densities in identical containers, and the average number of eggs laid per beetle was recorded. | Beetles per container | Eggs per…
Flour beetles were raised at different densities in identical containers, and the average number of eggs laid per beetle was recorded.
| Beetles per container | Eggs per beetle |
|---|---|
| 10 | 40 |
| 50 | 25 |
| 100 | 12 |
| 200 | 4 |
(a) Describe the trend. (b) Identify the type of limiting factor shown, and explain how it relates to carrying capacity. (c) Predict what would happen to the growth rate of the population as density keeps increasing.
Two rodent populations were counted at the same time each year for eight years. | Year | Population A | Population B | |---|---|---| | 1 | 480 | 300 | | 2 | 510 | 900 |…
Two rodent populations were counted at the same time each year for eight years.
| Year | Population A | Population B |
|---|---|---|
| 1 | 480 | 300 |
| 2 | 510 | 900 |
| 3 | 495 | 150 |
| 4 | 505 | 800 |
| 5 | 500 | 120 |
| 6 | 490 | 600 |
| 7 | 510 | 90 |
| 8 | 500 | 700 |
(a) Describe the pattern of each population. (b) Argue which population is more likely regulated by density-dependent factors and which by density-independent factors, such as rainfall. (c) Predict how a new predator that eats more prey when prey are abundant would affect each population, and justify your answer.
Using the models to manage populations
Wildlife managers use the logistic equation to decide how many animals can be harvested or removed. Because growth is fastest at $K/2$, a population there replaces the most individuals per year. Harvesting at that size takes the largest sustainable number.
Two cautions. Near $K$, growth is so slow that any harvest quickly drives the population down. And density-independent events, such as a severe storm, can cut the population suddenly, so managers build in a safety margin.
Back to St. Matthew Island: the reindeer had no predators and a lot of lichen, so they grew exponentially. They overshot the carrying capacity, ate the lichen supply, and then a harsh winter (a density-independent event) hit a population already short of food. The population crashed.
A sea otter population has $r_{max} = 0.3$ per year and a carrying capacity of $K = 2{,}000$. Wildlife managers want to allow a sustainable harvest. (a) Using the…
A sea otter population has $r_{max} = 0.3$ per year and a carrying capacity of $K = 2{,}000$. Wildlife managers want to allow a sustainable harvest. (a) Using the logistic equation $\dfrac{dN}{dt} = r_{max}N\left(\dfrac{K-N}{K}\right)$, calculate the growth rate at $N = 500$, $N = 1{,}000$, and $N = 1{,}500$. (b) Argue at which population size a harvest would be most sustainable, and explain why. (c) Explain what would happen if harvesting continued at a population near carrying capacity, and what a severe storm (density-independent event) would do to the harvest plan.
Summary
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Exponential growth ($\dfrac{dN}{dt} = r_{max}N$) gives a J-shaped curve with unlimited resources. Larger $N$ means faster growth.
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Logistic growth ($\dfrac{dN}{dt} = r_{max}N\dfrac{K-N}{K}$) gives an S-shaped curve that levels off at the carrying capacity $K$.
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Logistic growth is fastest at $N = K/2$ and is zero at $N = K$.
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Density-dependent factors (competition, disease, predation) get stronger with crowding and act as negative feedback.
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Density-independent factors (hurricanes, fires, freezes) act regardless of density.
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Managers harvest near $K/2$, where the growth rate is highest.