Population Growth Exponential and Logistic Models vs. Complex Reality 1 I. Exponential Population Growth A population is a group of individuals of the same species that live in the same area at the same time. In this activity you will analyze patterns of population growth and some practical implications. 1. Food contaminated with Salmonella bacteria can cause food poisoning, including diarrhea and other unpleasant symptoms. These symptoms develop when there are a lot of Salmonella bacteria in your intestines. Read the first column of this table and complete the second column. Here are some research findings about Salmonella food poisoning. You are more likely to get food poisoning if you eat contaminated food that is: raw or undercooked and has been kept at room temperature for several hours. Suggest a hypothesis that can explain these research findings. (Hint: Think about why this question is included in an activity on population growth.) There is a delay between when a person eats contaminated food and the beginning of diarrhea and other food poisoning symptoms. This delay is: shorter for people who consumed more Salmonella bacteria and longer for people who consumed fewer Salmonella bacteria. To better understand this example, you will analyze what happens if a single bacterium is placed in a flask that contains lots of food for bacteria. Every 30 minutes, each bacterium in the flask divides into two bacteria. Thus, the number of bacteria in the population doubles every 30 minutes. 2. How many bacteria do you think there will be by 5 hours after the single bacterium was placed in the flask (just guessing)? 1 By Dr. Ingrid Waldron, Dept. Biology, University of Pennsylvania, 2017. This Student Handout, a shorter version without equations, and Teacher Notes (with instructional suggestions and background biology) are available at http://serendip.brynmawr.edu/exchange/bioactivities/pop. 1
3. Complete this table to show how many bacteria there will be at each time if the number of bacteria doubles every 30 minutes. 1 hr. 2 hr. 3 hr. 4 hr. Time 0 min. 30 min. 1 hr. 2 hr. 3 hr. 4 hr. 30 min. 30 min. 30 min. 30 min. # Bacteria 1 2 5 hr. 4. Compare the calculated number of bacteria at 5 hours with your guess in question 2. 5. Graph the number of bacteria at each time. Connect the points to show the population growth curve. 6a. How long does it take for the population to increase from 1 bacterium to >500 bacteria? 6b. How long does 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 7. For these bacteria, population growth can be represented by the mathematical equation: ΔN = N, where N is the number of bacteria at the beginning of a 30-minute time interval ΔN (pronounced delta N) is the change in the number of bacteria from the beginning to the end of the 30-minute time interval Explain the biological reason why ΔN in a 30-minute interval is equal to N at the beginning of the 30-minute interval. 8a. Population growth for the bacteria in the flask is similar to population growth for Salmonella bacteria in contaminated food kept at room temperature. Explain why people are more likely to get food poisoning if they eat food that is raw or undercooked and has been kept at room temperature for several hours. 2
8b. Based on your analysis of exponential population growth, explain why people who have consumed fewer Salmonella bacteria experience a longer delay between eating contaminated food and the first symptoms of food poisoning. 9. For cottontail rabbits in the US, each year has a breeding season. A baby rabbit born in one year becomes a breeding adult by the beginning of the next breeding season. An adult female rabbit can have 3-4 litters of 4-5 baby rabbits each year. If a population begins with a pair of breeding adults and all the rabbits survive, how many rabbits do you think there would be at the end of six years (just guessing, without calculating)? To analyze rabbit population growth, we will assume that: Each adult female rabbit produces 20 baby rabbits each year. There is no mortality in the first six years, because food is abundant and there is no predation or disease. This figure shows the results of the first year of reproduction by a pair of adult rabbits. 10a. Use the information in the figure and the assumptions to complete this table. Year 1 2 Number of adults in this year (= number of rabbits at the beginning of the year = N) 2 Number of babies born in this year (ΔN) 10b. Explain why the number of babies born is larger in year 2 than in year 1. 10c. Your results in 10a should show that, for these rabbits, ΔN = 10 N. Give a biological interpretation of this equation. 3
Even though rabbits reproduce sexually and bacteria reproduce by cell division, both populations show exponential population growth. In exponential population growth, the growth rate is proportional to the size of the population, so population growth is increasingly rapid as population size increases. The general equation for exponential population growth is ΔN = R N, where: ΔN is the change in the number of individuals in a population from the beginning to the end of a time interval. R is the per capita rate of change in the size of the population for this time interval. N is the number in the population at the beginning of the time interval. 11. Although the general form of the equation for exponential population growth is the same for bacteria and rabbits, there are some major differences in population growth between these two organisms. Complete this table to describe the differences in the specific parameters of the exponential growth equation for bacteria vs. rabbits. R Time Interval Bacteria 1 30 minutes Rabbits The following table shows exponential population growth for the rabbit population, assuming maximum reproduction and no mortality. Year # Baby Rabbits Total # Rabbits # Breeding Adults produced during the at the end of the for this year = N breeding season = ΔN year = N + ΔN 1 2 20 22 2 22 220 242 3 242 2420 2662 4 2662 26620 29,282 5 29,282 292,820 322,102 6 322,102 3,221,020 3,543,122 The enormous potential for exponential growth of rabbit populations is illustrated by the Australian experience. European settlers released dozens of European rabbits in the early to mid-nineteenth century. By the mid-twentieth century, there were more than half a billion rabbits. The extremely large numbers of rabbits have caused extensive ecological damage. In contrast, this type of extreme increase in population size has not been observed for native rabbit populations in North America. 12. Complete this table to explore factors that can limit exponential population growth. Assumptions of the Exponential Population Growth Model for Rabbits There is no mortality in the first six years, because food is abundant and there is no predation or disease. Discuss whether this assumption is likely to be true for all rabbit populations, particularly as population size increases. Each adult female rabbit produces 20 baby rabbits each year. 4
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). As a population gets larger, there is increasing competition for resources. This results in increased mortality and/or decreased reproduction. Therefore, 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. This type of population growth is called logistic population growth 13a. In the figure, label the curve that shows logistic population growth with a carrying capacity = K. Label the other curve, which shows exponential population growth. 13b. Complete each of the following sentences with the best match from these three algebraic expressions: K K - N (K - N)/K The number of individuals that can live in the environment is. When there are N individuals in the population: the number of additional individuals that can live in the environment is and the fraction of carrying capacity that is still available for population growth is. Population density is the number of individuals per area of land or volume of water. Density-dependent factors have a stronger effect on population growth as population density increases. For example, for plants, competition for water, soil nutrients and sunlight increases as the number of plants in a given area increases. Thus, as population density increases, limited availability of water, soil nutrients and sunlight results in decreased plant survival and reproduction, which in turn results in slowed population growth. 14. Match each item in the top list with the best match or matches from the bottom list. both exponential population growth and logistic population growth models exponential population growth model logistic population growth model a. includes effects of density-dependent factors b. includes effects of limited resources, which result in a limited carrying capacity c. does not include effects of density-dependent factors; resources assumed to be unlimited d. slow population growth at the beginning because few individuals are reproducing 5
This table shows the equations for the exponential and logistic models of population growth. Exponential population growth Logistic population growth ΔN = R N where ΔN is the change in the number of individuals from the beginning to the end of a time interval N is the number of individuals at the beginning of the time interval R is the per capita rate of change in the size of the population (K N) ΔN = R N --------- K where ΔN and N are the same as for exponential growth R is the intrinsic per capita rate of change in the size of the population K is the carrying capacity of the environment 15. Complete the following table. Explain why logistic population growth: looks like exponential population growth when N is small. Provide an explanation based on the equations for exponential and logistic population growth. Provide a biological explanation. slows to 0 as N gets close to K. is most rapid when N = 0.5 K and slower when N = 0.1K or N = 0.9K This figure shows the trends in population density for a laboratory population of Paramecia. These single cell organisms ate bacteria. This population of Paramecia received the same amount of food each day. 16. Explain why population growth stopped after about 16 days. Include the term density-dependent factor in your explanation. 6
III. Mathematical Models vs. Complex Reality In order to make a manageable mathematical model, scientists need to make some simplifying assumptions. Often these simplifying assumptions are not accurate for real populations. If the simplifying assumptions are not accurate for the population being studied, then the model s predictions will differ from actual trends in population size. 17. This table illustrates these points for the exponential and logistic population growth models. Fill in the empty box in this table. Simplifying assumptions Situations in which these simplifying assumptions tend to be accurate What happens when the simplifying assumptions are not accurate Exponential Population Growth Model Resources are unlimited, so population growth does not affect an individual s probability of survival or rate of reproduction. When a species moves into a new environment After a population has been drastically reduced (e.g. by severe weather or human hunting) 18a. The dots show the trends in population size for a laboratory population of Daphnia (water fleas) that had a constant supply of food. Describe the difference between the observed trends in population size vs. the logistic population growth curve. Logistic Population Growth Model Carrying capacity is constant. As population size approaches carrying capacity, mortality increases and/or reproduction decreases quickly enough so that population growth slows promptly and population size does not exceed the carrying capacity. Often begins when a species moves into a new environment or after a population has been drastically reduced; followed by slowing population growth in a stable environment Changes in carrying capacity can cause corresponding changes in population size. If the population grows beyond the carrying capacity, then population size must eventually decrease. If the population is too big, it can damage the environment and reduce carrying capacity. 18b. What do you think is the reason that the observed trends in population size differed from the logistic population growth curve? (Hint: Think about which of the simplifying assumptions for the logistic population growth model might not have been valid for this population.) 7
19. This figure illustrates an exception to the assumption that carrying capacity is constant. Explain how seasonal changes in carrying capacity can account for the annual cycles in population size observed for this population of cottontail rabbits in Ohio. 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. However, by the late 1930s the large population of reindeer had drastically reduced the amount of lichen (which the reindeer depend on for winter food). The slowgrowing lichen had not significantly regenerated by 1950. 20a. Circle the part of the graph that shows approximately exponential population growth. 20b. Draw the expected population trend if population growth had followed the logistic population growth model and the carrying capacity of the island was 1000 reindeer. 20c. Explain how both of the simplifying assumptions of the logistic population growth model were not accurate for this population of reindeer. (The simplifying assumptions are shown in question 17.) Use this information to explain why the trends in reindeer population size did not match the predictions of the logistic population growth model. 21a. Trends in population size can look quite different, depending on which data are available for analysis. Explain how this general principle is illustrated by the trends in reindeer population size. How would your description of trends in reindeer population size differ if you only had data for 1910-1937 vs. if you considered the data for the whole time period shown? 21b. Many researchers only measure population size once a year. What important feature of population trends would be missed if the researchers who studied the cottontail rabbits in Ohio had only measured population size in July of each year? 8
The previous examples illustrate that, if the simplifying assumptions of the exponential or logistic population growth model are not valid for a population, then the population does not show the expected trend in population size. In contrast, these figures show examples where exponential or logistic population growth has been observed in natural populations. Each of these populations of seals was recovering from a drastic reduction in population size caused primarily by human hunting. (Number of pups is an index of total population size.) Our final example analyzes trends in population size for three species that live together in a marine biological community off the coast of Alaska. This figure shows trends in population size for these three species: sea otters (arrows) sea urchins ( ) kelp ( ). Kelp is a type of algae which is anchored to rocky areas near the coast and grows up to 45 m long. Kelp is eaten by sea urchins. Sea urchins are eaten by sea otters. 22a. See otter population size increased during the 1960s and remained high through the 1970s and early 1980s. What happened to the sea urchin population size during this time period? 22b. What happened to the kelp population size during this time period? 22c. What happened to the kelp population size when sea otter population size decreased during the late 1980s and the 1990s? 22d. Explain the biological processes that resulted in a relationship between sea otter population size and kelp population size. 9
23. Complete the following table to summarize what you have learned. Exponential Population Growth Model Logistic Population Growth Model What trends in population size are predicted by this model? Describe the types of situation where actual changes in population size tend to be similar to the predictions of this model. What are some reasons why growth of real populations often differ from the predictions of this model? 10