# Population Growth Exponential and Logistic Models vs. Complex Reality 1

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1 Population Growth Exponential and Logistic Models vs. Complex Reality 1 I. Exponential Population Growth 1a. Suppose a single bacterium is placed in a flask that contains lots of food for bacteria. In this flask, each bacterium grows and divides in two every 30 minutes. Therefore, 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 (just guessing)? 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 minutes bacteria by 1 hour bacteria by 1 hour and 30 minutes bacteria by 2 hours bacteria by 2 hours and 30 minutes bacteria by 3 hours bacteria by 3 hours and 30 minutes bacteria by 4 hours bacteria by 4 hours and 30 minutes bacteria by 5 hours 2. Plot the number of bacteria at each time; connect the points to show the population growth. Number of Bacteria Time (hours) 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. 1 By Dr. Ingrid Waldron, Dept. Biology, University of Pennsylvania, This Student Handout, the Teacher Notes, and an alternative version without equations are available at 1

2 4. 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. 5. A female cottontail rabbit begins reproducing by one year of age and typically has 3-4 litters of 4-5 baby rabbits each year. A cottontail rabbit can live up to eight 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)? 6. Calculate rabbit population growth over the first six years, based on these assumptions: The population starts with 1 female rabbit and 1 male rabbit, both born in the previous year. Each year has a breeding season, and each baby rabbit born in the previous year matures to become a breeding adult by the beginning of the next breeding season. Therefore, N (the number of rabbits in the population at the beginning of a year) = the number of breeding adults for the year. Each rabbit lives for eight years (maximum survival due to abundant food and no predation or disease), so there will be no deaths during the first six years. Once a female begins breeding, she produces 20 baby rabbits each year for the rest of her life. Half of the babies are female, so half of the population is female. Therefore, ΔN (the change in the number of rabbits from the beginning of the year to the end of the year) = the number of baby rabbits born that year = ½N * 20 = 10N. In other words, the increase in the number of rabbits each year equals the number of breeding females times the number of baby rabbits each female has in a year. N + ΔN at the end of a year will equal N at the beginning of the next year. 6a. Complete the following table. ear Y # Breeding Adults for this year = N # Baby Rabbits produced during the breeding season = ΔN Total # Rabbits at the end of the year = N + ΔN 6b. How close was your guess to the calculated number of rabbits by six years? 2

3 6c. Explain the biological reason why ΔN (the number of baby rabbits produced during the breeding season) is smallest in the first year and bigger in each successive year. As the size of the rabbit population increases, there is a proportionate increase in the growth of the rabbit population; thus, the rabbit population shows exponential growth. The enormous potential for exponential growth of rabbit populations is illustrated by the experience in Australia where 24 wild rabbits released in 1859 resulted in a population of 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. The general equation for exponential population growth is: ΔN = R N, where ΔN is the change in number in the population from the beginning to the end of a time interval N is the number in the population at the beginning of the time interval R is the per capita rate of change in the size of the population 7a. The equation shown above can be thought of as one type of model of exponential population growth, and the graph (on page 1) can be thought of as another type of model of exponential population growth. Comparing these two types of models, what is one advantage of a graph as a model of exponential population growth? 7b. What is one advantage of the equation as a model of exponential population growth? (Hint: Think about how you calculated the expected number of rabbits after six years.) 8. 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 6 do you think are probably not true for most rabbit populations? Which of these assumptions are probably increasingly unrealistic as population size increases? 3

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 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. Time 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 9. 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. (K N) ΔN = R N K where ΔN, N and R are the same as for exponential growth K = the carrying capacity of the environment Provide a biological explanation. Slows to 0 as N gets close to K. Is more rapid when N = 0.5 K and slower when N = 0.1K or N = 0.9K 4

5 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 will produce predictions that differ from actual trends in population size. These points are illustrated in this table and the examples that follow. 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 The per capita rate of change in population size is constant; this depends on abundant resources for the growing population. When a species moves into a new environment After a population has been drastically reduced (e.g. by severe weather or human hunting) May see logistic population growth Logistic Population Growth Model Carrying capacity is constant. As population size approaches carrying capacity, mortality increases and/or birth rates decrease so population growth slows promptly and the population 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. 10a. This figure shows the changes in population size for a laboratory population of Daphnia (water fleas) that had a constant supply of food. Why did the trends in population size differ from an exponential growth curve? 10b. What do you think is the reason that the 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 was probably not valid for this population.) 5

6 11a. This question includes two examples of exceptions to the assumption that carrying capacity is constant. First, 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. Jan A J O Jan A J O Jan A J O Jan A J (A = April; J = July; O = October) O 11b. 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 suggest a hypothesis to explain how wolves influenced the carrying capacity for songbirds. 12a. Some of the factors that influence changes in population size have density-dependent effects that is, the effects of these factors increase as population density increases (population density is the number of individuals per area of land or volume of water). Name some the factors we have considered that have density-dependent effects. 12b. Does the exponential model include density-dependent effects? yes no Does the logistic model include density-dependent effects? yes no Explain your reasoning. 12c. Some factors have density-independent effects on changes in population size. For example, freezing weather, tornadoes, floods and fires can drastically reduce population size, independent of initial population density. These types of density-independent effects are not included in the exponential or the logistic population growth models, so these models do not accurately predict changes in population size for a population that is strongly influenced by density-independent effects. Give an example to illustrate this. 6

7 13. 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. 13a. 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. Which model: Biological explanation: (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: , 2003) 13b. 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. Which model: Biological explanation: (Number of pups as an index of total population size; K = carrying capacity; from Hucke-Gaete et al., Polar Biology 27: , 2004) 13c. What is one hypothesis that could account for the difference in the population growth curves for these two populations of seals? 7

8 13d. 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 depend on for winter food). Which model: Biological explanation: 14. Complete the following table to summarize an advantage and a disadvantage for the exponential population growth model and for the logistic population growth model. Exponential Population Growth Model Logistic Population Growth Model One useful insight from this population growth model One reason why observed changes in population size often differ from the trends predicted by this model 8

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