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Population Growth – Exponential and Logistic Models vs. Complex Reality1

I. Exponential Population Growth1a. Suppose a single bacterium is placed in a flask filled with a broth that contains lots of nutrients. In this flask, each bacterium grows and divides in two every 30 minutes, so the number of bacteria in the population doubles every 30 minutes. How many bacteria do you think there will be by 5 hours after the single bacterium is placed in the flask?

1b. Complete the table to calculate how many bacteria there will be at each time.1 bacterium at the beginning = 0 minutes

bacteria by 30 minutesbacteria by 1 hourbacteria by 1 hour and 30 minutesbacteria by 2 hoursbacteria by 2 hours and 30 minutesbacteria by 3 hoursbacteria by 3 hours and 30 minutesbacteria by 4 hoursbacteria by 4 hours and 30 minutesbacteria by 5 hours

2. Plot the number of bacteria at each time; connect the points to show the population growth.!

Time

Name:________________

Number of Bacteria

2

3a. How long would it take for the population of bacteria to increase from 1 bacterium to 500 bacteria?

3b. How long would it take for the population to increase from 500 bacteria to 1000 bacteria?

Notice that, when a population doubles in each time interval, the number of bacteria in the population increases faster and faster as the population gets larger. This kind of population growth is called exponential population growth.

4. Rabbits can also show exponential population growth. A female cottontail rabbit begins reproducing by one year of age and typically has 3-4 litters of 4-5 baby rabbits each breeding season. A cottontail rabbit can live up to 8 years in captivity. If a population begins with a pair of breeding adults, and the rabbits have maximum reproduction and survival, how many rabbits do you think there would be after six years (just guessing, without calculating)?

5. To calculate rabbit population growth over the first six years, assume that:• A population starts with 1 female rabbit and 1 male rabbit, both one year old. • Each breeding female produces 20 baby rabbits per year; half of the babies are

female and half are male. • Each baby rabbit matures to become a breeding adult at the beginning of the

next breeding season. Thus, the number of breeding adults at the beginning of a year equals the total number of rabbits at the end of the previous year.

• These rabbits live in an environment with abundant food and no predation or disease, so all of the rabbits live out their lifespan (8 years) and continue to reproduce throughout their adult lives.

5a. Complete the following table, based on the above assumptions.

Year# Breeding Adults this

Year# Breeding Adults this

Year# Baby Rabbits

Produced during the Breeding Season

Total # Rabbits at the End of the YearYear

Males Females

# Baby Rabbits Produced during the

Breeding Season Total # Rabbits at the

End of the Year

1 1 1

23456

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5b. How close was your guess to the calculated number of rabbits by six years?

5c. Notice that the total number of rabbits at the end of each year is more than 10 times as large as the total at the end of the preceding year. What features of rabbit biology allow a rabbit population to increase more than tenfold in a year?

5d. Why is the number of baby rabbits produced smaller in the first year and bigger in each successive year?

Since the rate of increase in the rabbit population size increases as the size of the population increases, the rabbit population shows exponential growth. The enormous potential for exponential growth of rabbit populations is illustrated by the increase in Australia from 24 wild rabbits released in 1859 to more than half a billion rabbits by the 1920s. Notice that, even though rabbits reproduce sexually and bacteria reproduce by cell division, both populations show exponential population growth.

6. Extrapolating the exponential population growth of rabbits raises the question: Why isn't the world completely covered in rabbits? Which of the assumptions in question 5 do you think are probably not true for most rabbit populations? Which of these assumptions are probably increasingly unrealistic as population size increases?

Name:________________

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II. Logistic Population Growth

Exponential population growth cannot continue forever, since all organisms require resources to grow and reproduce, and the environment where a population is growing has a limited supply of resources (e.g. a limited supply of food or water). As a population gets larger, there is increasing competition for resources. This results in decreased reproduction and/or increased mortality, so the rate of population growth slows down. Eventually, the population will reach a maximum size which is called the carrying capacity of the environment; the carrying capacity depends on the amount of resources available in the environment. The second figure illustrates this type of logistic population growth with a carrying capacity (K).

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8. This figure shows the increase in population size for Paramecia that preyed on bacteria and yeast which in turn fed on oatmeal in a laboratory culture.

! !

Name:________________

Time

7. What are some resources that animals or plants will increasingly compete for as the population size approaches carrying capacity?

Days

What is a likely biological reason why population growth slowed and eventually stopped, even though the supply of food was maintained?

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9. When wolves recolonized a national park in Canada, this resulted in increased carrying capacity for songbirds. Researchers compared two areas in the park: an area around a town which wolves avoided vs. an adjacent area with many wolves. The researchers found that the area with many wolves had:

• greater mortality for elk and lower elk population density• lower consumption of willow leaves and branches, so the willows grew more• greater abundance and diversity of songbirds.

Use these research results to explain how wolves influenced the carrying capacity for songbirds.

10. For each of the following graphs: State whether the changes in population size shown in the graph follow:

• the exponential population growth model !• the logistic population growth model! !• a different pattern of population growth.

Give a biological explanation for the observed changes in population size; include any effects that may have resulted from changes in carrying capacity.

10a. This figure shows the growth of a population of gray seals on a Canadian island; this population was recovering from a drastic reduction in population size due to human hunting.

(Number of pups is an index of total population size; the different symbols represent the scientists' different methods of estimating the number of pups; from Bowen et al., ICES Journal of Marine Science 60:1265-74, 2003)

Which model:

Biological explanation:

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10b. This figure shows the growth of a population of Antarctic fur seals on islands near Antarctica; this population was also recovering from a drastic reduction in population size due to human hunting.

(Number of pups as an index of total population size; K = carrying capacity; from Hucke-Gaete et al., Polar Biology 27:304-311, 2004)

Which model:

Biological explanation:

10c. What is one hypothesis that could account for the difference in the population growth curves for these two populations of seals?

10d. This figure shows annual cycles in population size for a population of Eastern cottontail rabbits in Ohio.

Jan A J O Jan A J O Jan A J O Jan A J O (http://www.populationeducation.org/?option=com_content&task=view&id=38&Itemid=4)

Which model:

Biological explanation:

Name:________________

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10e. This figure shows changes in the size of a population that began with 25 reindeer on an island off the coast of Alaska. Initially, food was plentiful, but after several decades the large population of reindeer had drastically reduced the amount of lichen which the reindeer depended on for winter food.

http://go.hrw.com/resources/go_sc/bpe/HE0PE332.PDF

Which model:

Biological explanation:

Name:________________