Carrying Capacity and Demographics

Investigation Manual

Carrying Capacity and Demographics

BIOLOGY

Dry Lab

CARRYING CAPACITY AND DEMOGRAPHICS

Overview Modeling population growth is an important aspect of environ- mental and social sciences. In the first two activities, students simulate the population growth of an asexually reproducing organism without a carrying capacity and then of a bird popula- tion on an island with a set carrying capacity. In the third activity, they use birth and death dates to examine human demographics and survivorship curves. For the final activity, students use census data to visualize the age structure of world populations and gain insight into major historic events.

Outcomes • Compare logistic and exponential growth. • Examine the interactions between birth and death rates, and

how they affect population growth. • Apply the concept of carrying capacity. • Use demographics to make predictions about the growth of

national populations.

Time Requirements Preparation …………………………………………………………….. 5 minutes Activity 1: Simulation without Carrying Capacity ……….. 15 minutes Activity 2: Simulation with Carrying Capacity ……………. 45 minutes Activity 3: Cohort Analysis ……………………………………………2 hours Activity 4: Population Pyramids ………………………………. 30 minutes

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Table of Contents

2 Overview 2 Outcomes 2 Time Requirements 3 Background 6 Materials 6 Safety 6 Preparation 7 Activity 1 7 Activity 2 9 Activity 3 10 Activity 4 11 Disposal and Cleanup 13 Observations

Background Examining changes in populations over time is essential in many fields. Wildlife and fisheries managers need to make predictions about future population sizes to set acceptable hunting/catch limits. Economists want to know about human populations in different regions so they can make predictions about economic conditions. Demography is the study of factors that affect the rate at which populations, including human populations, grow.

Growth Rate Growth rate describes how a population is changing. It is affected by four main factors: birth, death, immigration (individuals moving into a population), and emigration (individuals moving out of a population). Mathematically, the growth rate (r) of a population can be represented as:

r = (births + immigration) – (deaths + emigration)

If increases due to births and immigration are greater than losses from death and emigration, then the growth rate is positive and the popula- tion is increasing. If the decreases from deaths and emigration are greater than increases from births and immigration, then the growth rate is negative and the population is shrinking. When the growth rate is zero, the population is holding steady.

Population Growth and Carrying Capacity In 1798, Thomas R. Malthus published a mathematical description of unlimited popu- lation growth. He argued that human popula- tions continue to grow until limited by famine, disease, poverty, or war. In mathematical terms, the current size of a population is equal to the

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previous size of the population multiplied by the growth rate:

N(t + 1) = Ntr where

N t = the number in the population at time t r = the growth rate constant

This formula reflects unlimited exponential growth if the value of r is greater than 1. If r is equal to 1, there is no change in the population size. If it is less than 1, the population declines to extinction. The formula depends on discrete intervals of time. To calculate growth over a continuous period, calculus is required.

In response to Malthus’s formula, other popu- lation scientists argued that wild populations frequently stay steady over time. They attributed a slowing of a population’s growth rate to a limitation of resources in the environment and suggested that population sustainability is due to a carrying capacity dependent on resources. Carrying capacity is the maximum sustainable size a population can reach within its environ- ment. Pierre Verhulst first based a formula on the presumption that, as a population approaches the carrying capacity of its environment, the rate of growth is slowed. That formula is now gener- ally presented as follows:

N(t + 1) = Ntr [(K – Nt)] where

N t = the number in the population at time t K = the carrying capacity

r = the growth rate constant

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K

CARRYING CAPACITY AND DEMOGRAPHICS

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This type of growth is called logistic growth. The factors that contribute to this type of growth rate can be summarized as birth, death, immigration, and emigration. Individuals can enter a population only by being born into it or moving into it. Likewise, they exit only by dying or moving out.

Population-limiting factors may be density dependent or density independent for carrying capacity. Among the density- dependent factors are decreased resource availability (e.g., food, water, and space), competition with similar species, disease and parasitism, and buildup of waste. The most frequent density-independent factors are natural disasters or extreme weather.

Population Demographics Like all organisms, humans are affected by resource quality and availability. A human population will grow until the carrying capacity of its environment is met. For instance, archeological evidence indicates that drought conditions contributed to the collapse of the ancient Ancestral Pueblo (Anasazi) empire in the 12th and 13th centuries. Similarly, disease, drought, and exhaustion of soil nutrients are commonly cited as explanations for the rapid decline in population that marked the end of the Classic Maya civilization in the 8th and 9th centuries. Today, scientists examine a wide range of characteristics, or demographics, to study population trends. Demographics are shaped in part by resources. Demographic data includes numbers of births and deaths, age, rates of emigration and immigration, and many other statistics.

Background continued

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One way that demographers evaluate a population is to generate a graph called a population pyramid. This analytical tool shows how the total population is subdivided into age brackets. A population pyramid can be used to predict future growth. Three population pyramids for 2016 are demonstrated in Figure 1, illustrating populations that are undergoing negative growth, rapid growth, and slow growth, respectively.

The population pyramid is a period analysis that works from a snapshot of the population at one time. This can be useful for understanding a current situation but does not allow for particularly accurate projections of the future population.

The alternative is cohort analysis, in which the population is broken into cohorts that are then followed over time. This allows for greater accuracy, because parameters such as birth, death, and migration rates can be age specific. The effects of any variations, such as a baby boom, can be tracked over time.

A cohort is a group of individuals who do something (e.g., are born) in the same time period. The researcher decides on age groupings that can be split enough to reflect important life stages but are not so numerous as to overwhelm the computational capacity. A cohort of individuals moves together through various age groups over time and is often subdivided by sex.

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The death rate is the number of individuals who die within the set time period divided by the number of individuals who belonged to the cohort at the beginning of the time period.

crude death rate = number of deaths total population

Likewise, the birth rate is the number of live births divided by the number of individuals. It can be hard to track marriages, divorces, deaths, and remarriages, so to keep from counting one child twice (for each parent), the birth rate is generally given as the number of births per each woman in the population.

crude birth rate = number of births total population

Survivorship curves (see Figure 2) show the attrition of a single cohort over the entire time that any members survive. The curves are generally set up as the number or percentage of surviving individuals versus age or time. Different organisms have different strategies for ensuring the next generation.

Figure 1.

–20,000,000 –10,000,000 0 10,000,000 20,000,000

–20,000,000 –10,000,000 0 10,000,000 20,000,000

–20,000,000 –10,000,000 0 10,000,000 20,000,000

Figure 2.

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Materials Needed but not supplied: • Paper, 6 sheets • Graph paper • Printer • Computer with internet access • Cup of 160 counters, such as dried beans, dry

rice grains, or pennies

Safety There are no safety concerns for this investigation.

Preparation Read through the activities completely.

K-strategists (named for the K in the carrying capacity equation) typically have fewer offspring but invest more energy in caring for the young. Elephants are an example of this strategy. Elephants give birth to only one calf at a time, and most have five years between calves. Calves stay under the protection of the mother and her herd until maturity at 10 to 15 years of age. Organisms of this strategy exhibit a Type I survivorship curve, having a slow decline at the beginning and middle of the curve (a low early mortality) followed by a rapid decline (most mortality among older individuals).

In the r strategy, organisms invest energy in producing many offspring in the hope that some will survive. Little energy is invested in the care for the offspring. Oysters, for example, spawn millions of eggs and sperm into a water column. Most of the eggs fail to be fertilized much less settle to grow into adult oysters. But the sheer number of offspring produced ensures the continuation of the species. Organisms that are r strategists usually have a rapid decline in their survivorship curve (a high mortality among the young) followed by a leveling off (a lower mortality for mature individuals). A Type III survivorship curve results from this kind of strategy.

Type II curves lie between these two extremes.

Background continued

8. Return all the counters to the cup. 9. Graph your results for all four values of

r. Put all four lines on the same graph for comparison. Be sure to label both axes and title your graph.

ACTIVITY 2

A Simulation with Carrying Capacity In this activity, you will model and document the changes in a population of birds over time. This population does not meet the assumptions made in Activity 1, so this investigation takes into account several additional factors. Birth Rate This model examines the bird population on a small island. Each nesting pair defends a number of territories. The more territories a pair can control, the more resources it has access to and the more chicks the pair can raise to adulthood. There are 32 territories on this island, each of which has the minimum space required to raise a single chick to adulthood. A given pair of birds can occupy from 1 to 4 of these territories and will have a corresponding number of offspring (1, 2, 3, or 4) each breeding season. As the population increases, competition will cause the ranges of the breeding pairs to shrink to the minimum size of 1 territory. Death Rate Initially, assume a constant death rate of 0.1. Then repeat the calculations using a death rate of 0.5. Immigration and Emigration Strong prevailing winds prevent immigration to the island. However, adults that cannot find a

ACTIVITY 1

A Simulation without Carrying Capacity

This activity simulates the growth of an asexually reproducing organism, such as yeast or bacteria. The growth rate can be thought of as the number of individuals dividing in a given time period.

1. Write “time = 0” on one sheet of paper. Write “time = 1” on the next, and continue through “time = 5” on the sixth sheet.

2. Count out 5 counters, and put them on the paper labeled “time = 0.” This is your starting population. Record the number in Data Table 1.

3. Assume that the population growth rate (r) is 2.0. In other words, at each time interval, the population is double the size of the population in the previous period.

4. Calculate how many counters should be in the next time interval, and count them out onto the paper labeled “time = 1.” Record the new population number in Data Table 1. This is one iteration of this model. An iteration is one complete set of steps in a repeating sequence of those steps.

5. Repeat Step 4 until you have completed calculating and recording data through “time = 5.”

6. Return all the counters to the cup. 7. Repeat Steps 2 to 6 three more times,

assuming the following population growth rates: r = 1.5, r = 0.7, and r = 1.0.

If your calculation results in a decimal, round down to the next whole number regardless of the value. In other words, if you calculate 4.7, use 4 counters.

ACTIVITY

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ACTIVITY 2 continued

ACTIVITY

A breeding pair can control up to four territories. Therefore, each breeding pair can have from 1 to 4 offspring. Example: If a breeding pair controls four territories, the pair will have four offspring.

c. Birth rate per female: Divide the number of offspring by the number of females. (In this instance, the number of females is equal to the number of breeding pairs plus half the number of helpers.)

d. Number of deaths: Multiply the death rate by the number of birds in the population at the beginning of the time interval (N). Round down to the nearest whole number.

e. Number driven off (emigration): Add the starting number of birds (N) and the number of offspring produced for that time interval. Then subtract the number of deaths as determined by the death rate. The resulting number is the number driven off. If the number is less than or equal to the carrying capacity (96), write zero as the number driven off.

f. Number in the population (to begin the next iteration): Add the starting population and offspring births, and then subtract the number of deaths and number of birds driven off. Record the result as the number in the population (N) at the beginning of the next time interval.

g. Population growth rate: Divide the number in the population for the next time interval (e.g., time interval 1) by the number in the population for the current time interval (e.g., time interval 0).

4. Use the counters to help you visualize what is happening during each iteration. This will help

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place in one of the territories must emigrate to look for other places to live. Carrying Capacity Each territory can support 3 birds—a breeding pair and a helper—and still produce 1 offspring. (A helper typically is an offspring from the previous breeding season that cannot get a territory of its own and stays to help its parents raise the next brood.) The carrying capacity of the island is 96 birds: 32 breeding males, 32 breeding females, and 32 helpers. If the bird population exceeds the island’s carrying capacity, the additional birds are forced to emigrate. 1. Print a copy of the Bird Island map (Figure 3,

page 12), and obtain a cup of 100 counters. 2. Place 4 counters on the map. Separate

the counters to create two breeding pairs, and then place each pair in the center of four small territories. These four territories represent the breeding range controlled by the pair. In Data Table 2A, in the time interval row for zero, write “4” as the “Number in the population (N) at Beginning of Interval” column. For the first simulation, use a death rate of 0.1.

3. Perform the following calculations, and record the results in appropriate columns in Data Table 2A.

a. Number of breeding pairs: Divide the population by 2. Single birds cannot reproduce. Remember that the island accommodates only 32 breeding pairs.

b. Number of offspring: Each breeding pair will have 1 offspring per territory in the breeding range controlled by that pair.

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you keep track of how many territories each breeding pair controls. For each time interval:

a. Add counters to the island to represent the number of offspring produced in that interval.

b. Determine the total number of deaths and birds driven off from the time interval, and remove those counters from the map. Return the counters to the cup.

c. Redistribute the counters to represent breeding pairs. Breeding pairs prefer to spread out. If possible, spread out the pairs to allow them to control a range of multiple territories, up to a maximum of 4 territories. Note that once the population increases to a certain size, this will no longer be possible, and each breeding pair will lose territories until it controls only 1 territory. (As the population grows, the range controlled by each breeding pair will decrease.)

d. Repeat the iterations until the population of birds remains unchanged for 3 consecutive time intervals.

Each territory can support a breeding pair and a helper bird—a maximum of 3 birds per territory. At carrying capacity, each territory will support a breeding pair and a helper.

5. Repeat the simulation using a death rate of 0.5. Assume the carrying capacity of the island remains at 96. Record your data in Data Table 2B.

6. Graph the results of the two simulations, and note the differences over time between a

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death rate of 0.1 and a death rate of 0.5. Plot both lines on the same graph for comparison. Remember to label your axes and title your graph.

Activity 3

A Cohort Analysis 1. For this activity, you will obtain information

on life expectancies and mortality rates. Your instructor will assign you one of the following sources from which to collect data:

a. Visit a cemetery, and record the data from tombstones.

It is best to visit community cemeteries. Using specialized cemeteries, such as Arlington National Cemetery, will skew the demographic data.

b. Work with names taken from online databases of cemeteries.

2. Choose two groups to compare. You could choose two different birth cohorts (a group of individuals born within a defined timespan, usually 10 to 25 years). Alternatively, you may choose to look at males and females in the same birth cohort. All years of birth in this activity should be before 1910. If you are using online databases, you also may choose to compare two different regions of the country or world.

3. Collect data from your assigned source. A minimum of 50 individuals in each group is recommended, but more will yield better results. Record the year of birth, year of death, and gender in Data Table 3. You will

need to print multiple copies of Data Table 3 to accommodate your sample size.

4. Calculate the age at death by subtracting the birth year from the year of death. Record this in Data Table 3.

5. For each cohort being examined, calculate the mean (average) lifespan for your sample data. Record it in Data Table 3 as the average age of death.

6. For each cohort, calculate the number of individuals who died in each 10-year period after the initial birth year. For example, if the cohort is defined as individuals born between 1900 and 1910, determine the number of individuals who died before 1910, the number of individuals who died from 1910 through 1919, the number of individuals who died from 1920 through 1929, etc. Record this in Data Table 4 in the second column.

7. Calculate the crude death rate of each cohort. a. Divide the number of deaths by the total

number of individuals at the start of the cohort.

b. Multiply the result by 1,000 to standardize the mortality rate to 1,000 individuals. This is the crude death rate. Record the death rate in Data Table 4.

8. Calculate the survivorship of each cohort. a. Subtract the number of deaths from the

total number of individuals in the cohort. This is the number of survivors. Record this number in Data Table 4.

b. To standardize the survivorship to 1,000 individuals, divide the number of survivors for the 10-year time period by the total

ACTIVITY

number of individuals at the start of the cohort, and record the quotient in Data Table 4. The S1000 is a standard measurement of survivorship.

c. Graph the survivorship vs. time for each cohort.

Activity 4

A Population Pyramids The United States Census Bureau keeps a comprehensive database of demographic data from around the world. Summaries of the International Data Base can be found on the International Programs page. (https:// www.census.gov/data-tools/demo/idb/ informationGateway.php)

Comparing Countries 1. From the “Select Report” drop-down menu,

select “Population Pyramid Graph.” 2. Choose the current year from the list of years. 3. From the list of countries or areas, select

“India,” “Italy,” “United States,” and “Zimbabwe.” (To select multiple countries, hold down the Ctrl key on Windows machines or the command key on Apple computers.)

4. Click “Submit.” 5. Sketch the graph for each country in Data

Table 5. Be sure to note the scale on each. 6. Write your observations about each graph.

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ACTIVITY 3 continued

https://www.census.gov/data-tools/demo/idb/informationGateway.php

https://www.census.gov/data-tools/demo/idb/informationGateway.php

https://www.census.gov/data-tools/demo/idb/informationGateway.php

https://www.census.gov/data-tools/demo/idb/informationGateway.php

Disposal and Cleanup There is no disposal or cleanup necessary for this investigation.

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Following a Population through Time 1. Click the “Search” tab at the top of the

section to return to the search menu. 2. From the year menu, select the current year.

Holding down the shift key, select the prior 20 years.

3. From the list of countries, select “United States.”

4. Click “Submit.” 5. You can either scroll through the results, or

you can click the play button at the top of the first graph next to the heading “Animation Controls.”

6. Make observations of how the population has changed over the past 20 years.

7. Repeat this procedure for Russia, selecting 1990 to 2010 as the year range.

8. Note the large dip in those who were in their late 40s in the early 1990s. In what other countries might you expect to find a similar pattern? Hint: Consider what major world events occurred during the lifespans of this cohort. Make a prediction, and test your hypothesis.

ACTIVITY

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Figure 3.

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Carrying Capacity (K) = 96 Death Rate = 0.1 Time

Interval (t) Number in Population

(N) at Beginning of Interval

Number of Breeding

Pairs

Number of Offspring

Birth Rate Number of Deaths

Number Driven Off

Population Growth Rate (r)

0 1 2 3 4 5 6 7 8 9 10

Data Table 2A.

OBSERVATIONS

Number in Population (N)

Time Interval (t) r = 2.0 r = 1.5 r = 0.7 r = 1.0

0

1

2

3

4

5

Data Table 1.

OBSERVATIONS continued

ACTIVITY

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Carrying Capacity (K) = 96 Death Rate = 0.5 Time

Interval (t) Number in Population

(N) at Beginning of Interval

Number of Breeding

Pairs

Number of Offspring

Birth Rate Number of Deaths

Number Driven Off

Population Growth Rate (r)

0 1 2 3 4 5 6 7 8 9 10

Data Table 2B.

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Birth Year Death Year Age at Death Gender

Average age at death: ____________

Data Table 3.

OBSERVATIONS continued

ACTIVITY

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Years Number of Deaths Crude Death

Rate Number of Survivors

Survivorship per 1,000 (S1000)

0–10 11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90

91–100 >100

Data Table 4.

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Country Graph Observation

India

Italy

United States

Zimbabwe

Data Table 5.

NOTES

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BIOLOGY Carrying Capacity and Demographics

Investigation Manual

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Carrying Capacity and Demographics

Table of Contents

Overview

Outcomes

Time Requirements

Key

Background

Growth Rate

Population Growth and Carrying Capacity

Population Demographics

Materials

Safety

Preparation

ACTIVITY 1

A Simulation without Carrying Capacity

ACTIVITY 2

A Simulation with Carrying Capacity

Birth Rate

Death Rate

Immigration and Emigration

Carrying Capacity

Activity 3

A Cohort Analysis

Activity 4

A Population Pyramids

Comparing Countries

Following a Population through Time

Disposal and Cleanup

OBSERVATIONS

NOTES

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