Unit 3 · Topic 3.5 Beta

Population Growth and Resource Availability

With unlimited resources, a population grows exponentially (a J-curve), by a constant percent each period.

Practice 5: Data AnalysisPractice 6: Mathematical Routines

Part 1 · Hook

Why this matters

A few rabbits released in Australia in 1859 had no foxes or diseases to hold them back and a whole continent of grass ahead of them. Within a few decades there were hundreds of millions. In a fenced field, the same rabbits would grow fast at first and then level off as grass and space ran short. Same animal, two growth curves: the J and the S.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is carrying capacity?

  1. The largest population an environment can support over time
  2. The fastest a population can grow
  3. The number of species in an area
  4. The age when a species first breeds
Show the answer

Carrying capacity (K) is set by an environment’s resources.

  • Correct: The largest population an environment can support over time:
  • The fastest a population can grow:
  • The number of species in an area:
  • The age when a species first breeds:

2. A population grows 7% a year. About how long does it take to double, by the rule of 70?

  1. 10 years
  2. 7 years
  3. 70 years
  4. 0.1 years
Show the answer

70 divided by the growth rate in percent: 70 / 7 = 10 years.

  • Correct: 10 years:
  • 7 years:
  • 70 years:
  • 0.1 years:

Part 4 · See it

See it first

Two growth curves from the same start of 20 individuals. The exponential J-curve rises faster and faster and leaves the graph above 1,400 at about year 17. The logistic S-curve grows fastest at half the carrying capacity, 500, and levels off at the carrying capacity of 1,000 (dashed line).
Exponential growth (J-curve) with unlimited resources, and logistic growth (S-curve) that levels off at the carrying capacity. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Resources are plentiful and nothing limits a populationit grows by a constant percent each year, so it adds more individuals each year: exponential growth, a J-curve
  2. The population grows and resources are shared among more individualslimiting factors such as food, space and disease become stronger
  3. Births fall and deaths rise as the population nears carrying capacitygrowth slows and the population levels off at K: logistic growth, an S-curve

Part 6 · Key ideas

Key ideas

  • Exponential growth: with unlimited resources, a population grows by a constant percent each period, so it adds more individuals each time. The graph is a J-curve.
  • Logistic growth: growth is fast at first, slows as resources run short, and levels off at the carrying capacity. The graph is an S-curve; growth is fastest at about half of K.
  • A limiting factor is any resource or condition that keeps a population from growing: food, water, space, light, nutrients, predators, disease.
  • Density-dependent factors grow stronger as a population gets denser (food shortage, disease, competition, predation). Density-independent factors hit a population whatever its size (a flood, a frost, a fire).

Part 7 · Misconception

A common mistake

The wrong idea: A population growing exponentially adds the same number of individuals every year.

What actually happens: It grows by the same percent every year, so the number added grows each year: 2% of 1,000 is 20, but 2% of 2,000 is 40.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Data

Yeast growing in a flask

Yeast cells per milliliter in a flask of sugar solution, counted every 2 hours (invented data for practice)
Time (hours)Yeast cells per mL (thousands)
010
220
440
680
8150
10240
12300
14320
16325

1. During which period does the yeast population double every 2 hours?

  1. From 0 to 6 hours
  2. From 8 to 12 hours
  3. From 12 to 16 hours
  4. Over the whole 16 hours
Show the answer

Steady doubling is the sign of exponential growth.

  • Correct: From 0 to 6 hours: Right: 10, 20, 40, 80 thousand cells per mL, doubling each 2 hours.
  • From 8 to 12 hours: From 150 to 240 to 300, growth is slowing, not doubling.
  • From 12 to 16 hours: The population is leveling off: 300, 320, 325.
  • Over the whole 16 hours: Doubling stops after about 6 hours.

2. Which statement best describes the shape of the whole yeast growth curve?

  1. A J-curve that rises faster and faster to the end
  2. An S-curve leveling off near 325 thousand cells per mL
  3. A straight line rising at a steady rate
  4. A curve that rises and then crashes to zero
Show the answer

Rapid growth that slows and levels off is logistic.

  • A J-curve that rises faster and faster to the end: After 8 hours growth slows, so it is not a J-curve throughout.
  • Correct: An S-curve leveling off near 325 thousand cells per mL: Right: fast early growth, then slowing, then a plateau: logistic growth.
  • A straight line rising at a steady rate: Growth is not steady: it speeds up, then slows.
  • A curve that rises and then crashes to zero: The population levels off; it does not crash in these 16 hours.

3. Which explanation best accounts for the yeast population leveling off after 12 hours?

  1. The yeast cells stop dividing once they are 12 hours old
  2. The flask gets colder each hour of the experiment
  3. Sugar runs low and wastes build up
  4. Yeast stop growing at a set 300 thousand cells per mL
Show the answer

A population levels off when limiting factors strengthen as it grows.

  • The yeast cells stop dividing once they are 12 hours old: Cells keep dividing when there is food; age is not the limit.
  • The flask gets colder each hour of the experiment: Nothing suggests the temperature changed.
  • Correct: Sugar runs low and wastes build up: Right: density-dependent limits (food and waste) slow growth near carrying capacity.
  • Yeast stop growing at a set 300 thousand cells per mL: The level depends on the flask’s resources, not a fixed number.

Model or diagram

Two growth curves

Two growth curves from the same start of 20 individuals. The exponential J-curve rises faster and faster and leaves the graph above 1,400 at about year 17. The logistic S-curve grows fastest at half the carrying capacity, 500, and levels off at the carrying capacity of 1,000 (dashed line).
LevlPrep original diagram.

4. On the figure, at about what population size does the logistic curve grow fastest?

  1. About 1,000, at the carrying capacity
  2. About 20, at the start
  3. About 1,400, at the top of the graph
  4. About 500, half the carrying capacity
Show the answer

Logistic growth is fastest at about half of K.

  • About 1,000, at the carrying capacity: At K the curve is flat; growth has stopped.
  • About 20, at the start: At the start the curve is low and flat; few individuals means few added.
  • About 1,400, at the top of the graph: The logistic curve never reaches 1,400.
  • Correct: About 500, half the carrying capacity: Right: the logistic curve is steepest at about K/2, marked on the figure.

5. Both curves start at 20 and match for the first few years. Why do they separate later?

  1. Limits slow the S-curve near K
  2. The J-curve species has a lower biotic potential
  3. The S-curve species dies out after 15 years
  4. The J-curve stops growing once it reaches the carrying capacity
Show the answer

When a population is small, limits barely matter, so logistic growth looks exponential.

  • Correct: Limits slow the S-curve near K: Right: limiting factors bend the logistic curve; the exponential curve assumes no limits.
  • The J-curve species has a lower biotic potential: Both start with the same growth; the J-curve is not slower.
  • The S-curve species dies out after 15 years: The S-curve levels off near 1,000; it does not die out.
  • The J-curve stops growing once it reaches the carrying capacity: The J-curve passes 1,000 and keeps rising.

6. A herd of 400 elk lives in a valley of 25 square kilometers. What is its population density?

Type a number, then its unit (elk/km^2). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

Number over area: 400elk25km2=16\frac{400\,\text{elk}}{25\,\text{km}^2} = 16 elk per km².

  • Answer: 16 elk/km^2

7. Which factor is density-independent?

  1. A disease that spreads faster in crowded herds
  2. Competition among birds for the few nesting sites
  3. A late frost that kills plants
  4. Predators that hunt more where prey is dense
Show the answer

Density-independent factors act no matter how many individuals there are.

  • A disease that spreads faster in crowded herds: Spread through crowding makes disease density-dependent.
  • Competition among birds for the few nesting sites: Competition gets stronger as density rises: density-dependent.
  • Correct: A late frost that kills plants: Right: a frost acts on the population regardless of its density.
  • Predators that hunt more where prey is dense: Predation that tracks prey density is density-dependent.

Part 9 · Summary

Summary

With unlimited resources, a population grows exponentially (a J-curve), by a constant percent each period. In real environments, limiting factors slow growth as the population nears carrying capacity, giving logistic growth (an S-curve). Density-dependent factors such as food shortage, disease and competition strengthen as a population grows; density-independent factors such as floods and frosts act whatever its size.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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