Population Dynamics Simulation PDF

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INTRODUCTION Merienne Moussa Wajdowicz In the Population Dynamics Click & Learn, you’ll explore a powerful tool for learning about populations: mathematical models. As you’ll see by doing the Click & Learn and this activity, you can use models to answer questions, solve problems, and make predictions about all kinds of populations — from bacteria in your body to wildlife across the world, to our own human populations. PART 1: Exploring the Exponential Growth Model Open the “Exponential growth model” tab and read the “Introduction” section. 1. The end of the “Introduction” describes how you could use a continuous-time, exponential growth model to simulate an E. coli population growing in a lab. Describe another specific population and situation that you could simulate with this type of model. For the time model, it would pick a population that will be changing, for the time around a year due to births or deaths. For growth, it would take food, space, or density. 2. Complete the following table to explain the biological meanings of the symbols in the exponential growth model. For each explanation, give a specific example using the population you described above. Symbol N t dN/dt r N0

Biological Meaning Specific Example It is the numbers of individuals in a How many plants are in the forest in the given population time The time How much time did it take to grow a plant It is the rate of the population size How quickly does the plant died and are born The possible growth in a population, per How the plants can gain more of them when individual it’s growing more like the normal The population size initial How many plants were at the beginning of the experiment

3. Both dN/dt and r are types of growth rates. What are the differences between them? dN/dt changes over time, r stays the same. dN/dt depends on the time, r does not depend on the time. dN/dt measured overall the population, r is measured per individual. 4. No units are shown for the numbers in the “Settings” section. This is because each of these numbers can have many possible units. Give an example of possible units for each of the following: a. N0 bacterias, animals, individuals, population, etc. b. t minutes, years, seconds, hours, months, etc. c. r, using the units for t you gave above second: individuals per second. minute: animals per minute. hours: bacterias per hour. years and months: population per time.

5. Examine both Plot 1 and Plot 2. a. In Plot 1, what variables do the x- and y-axes represent? The x-axis represents the time that it is t. The y-axis represents the population size that it is N. b. In Plot 2, what variables do the x- and y-axes represent? The y-axes represent the population growth the is dN/dt. The x-axes represent the population size that it is N 6. Set N0 = 50, r = 0.5, and t = 5. a. What is the population size at this point? The population size= N= 609 b. What is the population growth rate at this point? The population growth rate= dN/dt= 304.56 7. Set r = 0.1, then gradually increase r by clicking the up-arrow to the right of the number. You may need to hover over the number to see the arrow. a. Examine Plot 1. As you increase r, what happens to the curve of population size over time? When r increases, the curve becomes excessive quickly, and this shows that the size of the population increases rapidly. b. Examine Plot 2. As you increase r, what happens to the curve of population growth rate vs. population size? The curve will be always in a straight line, which means that ht population of growth increase with the population size. But the r increase and it becomes excessive that it indicates the population growth is increasing rapidly with the N. 8. Set r = 0.5 and N0 = 5, then gradually increase N0 by clicking the up-arrow to the right of the number. a. Examine Plot 1. How does the curve of population size over time change if you start with a smaller number of individuals (e.g., N0 = 5) compared to a larger number of individuals (e.g., N0 = 100)? When the population is with the numbers of the individuals that are small. The population over time increases rapidly over time, and this means that the population is growing faster. When the population is with a large number of individuals, the curve will rise quickly, and it means that the population is growing quickly from the start. b. Examine Plot 2. How does the curve of population growth rate vs. population size change if you start with a smaller number of individuals compared to a larger number? The growth will not change, because it does not change when the initial population size it is changing. And it means that the initial population size does not affect the relationship between the population size and the population growth size. 9. List one combination of values for r and N0 that produces each of the following patterns for population size over time. (There are many possible answers.) Use a time range with a “Min” of 0 and a “Max” of 10. Pattern A long period of what appears to be almost no growth. (The curve in Plot 1 looks almost flat.)

Value of r Small 0.2

Value of N0 Small 1

A long period of slow but clearly accelerating growth. (The curve in Plot 1 starts to become steeper at the end.) Extremely fast growth from the very beginning.

Intermediate 0.5

Small 1

Large 1

Large 100

PART 3: Exploring the Logistic Growth Model Go to the “Logistic growth model” section and read the introduction. 10. Summarize the main differences between the exponential and logistic growth models. The logistic growth is what the model describes in the population that is limited in the resource. the limits to growth and if it grows more slowly as it gets larger. Exponential growth describes the population that is with an unlimited of resources, and it keeps growing faster and bigger during time. 11. Explain what the carrying capacity (K) is in your own words. The carrying capacity is the size bigger than the population in an environment and it supports how long the species are.

Proceed to the “Simulator” section for the logistic growth model. 12. Set N0 = 1, r = 0.6, and K = 1000. Also set the “Max” value for t on the x-axis of Plot 1 to 25. a. Examine Plot 1. What happens to the population size over time? The population will be growing in size until it approaches its carrying capacity. b. Examine Plot 2. For what values of N is the population growth rate almost zero (for example, 0.01 or lower)? The population growth will be almost zero for the N that it is close to the 0 or close to the carrying capacity. c. Set N0 = 1500. What happens to the population size overtime now? For what values of N is the population growth rate almost zero? The population will be decreasing in size until it reaches the carrying capacity. The population growth rate is almost zero for N that are close to carrying the capacity.

d. In general, for what values of N and K is the population growth rate almost zero? In general, the population growth rate is almost zero for N that are close to carrying the capacity. 13. Set N0 = 1 again. Gradually increase r by clicking the up-arrow on its box. a. Examine Plot 1. As you increase r, what happens to the curve of population size over time? The population increases quickly in the beginning and it will be reaching the carrying capacity quickly. b. Examine Plot 2. As you increase r, what happens to the curve of population growth rate vs. population size? (Hint: Pay attention to the numbers on the y-axis of Plot 2.) The population growth will be low when N is close to the 0 or the carrying capacity, but it is between the two extremes. When the r increases in the y-axis and this indicates that the peak was getting tallet.

PART 4: Waterbuck Africa is home to many different kinds of animals, including large antelope called waterbuck that live near lakes and rivers. In certain areas, waterbuck populations are declining due to hunting and habitat loss. 1. How could we use mathematical models to help waterbuck and other wildlife? It could use the wildlife population for the future, and see if there is a population that is in danger. Let’s investigate the waterbuck population in Gorongosa National Park, Mozambique. In the 1970s and 1980s, Mozambique experienced an intense civil war, and most of the waterbuck were killed to provide food and money for soldiers. After the war ended in 1992, many people worked together to rebuild the park. Scientists developed mathematical models to better understand how the park’s waterbuck population recovered afterward, and to help make decisions about managing this population in the future. 2. What are the advantages of using a mathematical model to study a population rather than just observing the population? The models it will be preferable when observing the population directly would be hard because the animals are difficult to find, and it can be costly because the animals require the time and the types of equipment. The models can also be used to find the scenarios and the conditions and the thing that has not been tested yet or the predictions about the things that are happening. The population’s maximum per capita growth rate (r) was estimated as the difference between its per capita birth rate (b), the number of births per individual per unit time, and its per capita death rate (d), the number of deaths per individual per unit time: 𝑟 =𝑏–𝑑

3. At the start of the recovery period, the waterbuck population contained only 140 individuals. The population had 0.67 births per year per individual and 0.06 deaths per year per individual. a. What is the maximum per capita growth rate (r) for this population? Include units in your answer. 0.61 per year b. What is the initial population size (N0) for this population? Include units in your answer. 140 waterbucks Go to the “Simulator” section under the “Exponential growth model” tab in the Population Dynamics Click & Learn. Fill in the simulator settings based on your answers above. (Note: The simulator doesn’t show units for times or rates because many units are possible. In these examples, we’ll use “years” as our unit for time and “per year” as our units for per capita rates.) 4. Using the simulator, fill in the following table with the population size (N) and population growth rate (dN/dt) at different time points (t, measured in years). Time (t) Population size (N)

5 2,956

Population growth rate (dN/dt)

1,803.25

10 64,420

15 1,318,022

20 27,830,481

25 587,650,195

38,076.25

803,993,21

16,976,593.51

358,466,619.03

5. Based on this model, how will the waterbuck population grow over time? Will the population ever stop growing or get smaller?

In the model, the waterbuck over time grows faster and bigger. It does not stop or neither gets smaller. 6. Do you think this model reflects how the waterbuck population will grow in real life? Why or why not? The population is unlikely to grow forever like in the model. It may be something to limit the growth of the population. The population could run out of space or food or it could gt a disease. 7. Imagine that a decrease in the number of predators lowered the per capita death rate of the waterbuck to 0.04 deaths per year per individual. a. What would be the new maximum per capita growth rate (r) for the waterbuck population? 0.63 per year b. What would be the population size (N) after 20 years (t = 20)? Use the same N0 as in Question 3. 41,518,199 waterbuck

We originally estimated r as the difference between the per capita birth rate (b) and the per capita death rate (d). However, r is also affected by other processes, such as immigration (movement of individuals into a population) and emigration (movement of individuals out of a population). Let i represent the per capita immigration rate and m represent the per capita emigration rate. The equation for r can be updated to: 𝑟 = (𝑏 − 𝑑) + (𝑖 − 𝑚) 8. Imagine that new waterbuck immigrate into the park at a rate of 0.25 per year. Assume that there are no emigrations and that the rest of the population parameters are the same as in Question 3. a. What would be the population size after 20 years (t = 20)? 4,130,409,628 waterbuck b. How does the size of the population with immigration (your answer to Part A) compare to the size of the population without immigration (your result for t = 20 in Table 1)? The population size that is with immigration is bigger than the population without one. The small changes can have different impacts on the exponential growth model. PART 5: Kudu Another type of African antelope is kudu. Like waterbuck, many kudu has lost their habitat due to human activities. Male kudu are also hunted for their large spiraled horns, which are taken as trophies. As with waterbuck, developing population models for kudu can help us learn more about them. Most populations, including those of the waterbuck and kudu, cannot grow forever. They are limited by factors such as food or space, which keep a population from getting too large. 9. Besides food and space, what are two other factors that could limit the size of a population? The increased rate of diseases and parasitism when the population is large. And it could be to by loss of pollution or natural disasters. One model that includes the effect of limiting factors is the logistic growth model, which is described in the “Logistic growth model” section of the Population Dynamics Click & Learn. In this model, a population has an

upper limit to its growth called the carrying capacity (K), which is the largest size of a population that the environment can support in the long run. Imagine a national park with an initial population of 10 kudu, which has a maximum per capita growth rate of 0.26 per year. The park can support a maximum of 100 kudu in the long run. 10. What are the values of K, r, and N0 for this kudu population? N= 10 kudu, k= 100 kudu, r= 0.26 per year Go to the “Simulator” section under the “Logistic growth model” tab in the Population Dynamics Click & Learn. Fill in the simulator settings based on your answers above. 11. Based on this model, how many years will it take the kudu population to reach the carrying capacity? (Hint: You may want to change the “Max” values for the axes on Plot 1 to get a better look at the curve.) Around 29 years 12. What will happen to the population growth rate (dN/dt) as the population size (N) gets closer and closer to the carrying capacity? The growth will increase until the population reaches half of the carrying capacity that will be 50. The growth will decrease to 0 when the population gets closer to the carrying capacity that will be 100. 13. Imagine that more land is added to the park, allowing it to support up to 250 kudu. How will the size of the kudu population change once this land is added? The population will be growing until it reaches its carrying capacity that it will be 250 kudu. 14. Reset the model to the values you determined in Question 10. Now imagine that trophy hunters start killing kudu in the park, which decreases their maximum per capita growth rate to 0.15 per year. How would this impact the kudu’s population size over time? (Hint: Look at how many years it will take the population to reach its carrying capacity now.) The population will grow slowly after the hunter kills the kudu. The population will reach their carrying capacity in about 29 years without the hunter. The carrying capacity will not be reached in about 50 years with the hunter. PART 3: Wildebeest The last type of antelope we’ll investigate is the wildebeest, which are found in eastern and southern Africa. Wildebeest live in giant herds that can contain over a million individuals! The wildebeest herd in the Serengeti region of Tanzania is one of the biggest populations of large herbivores in the world. Before the 1960s, wildebeest and many other hoofed mammals in the Serengeti were killed by rinderpest, a virus related to the measles virus. In 1960, a campaign began to vaccinate domestic cattle, which were a major source of the virus. Over time, the campaign eliminated rinderpest and allowed many animal populations to recover. Figure 4 shows the population sizes of two animals, wildebeest, and zebras, before and after the rinderpest vaccination campaign.

Figure 1. Wildebeest and zebra populations in the Serengeti from the 1950s to 2010. 15. Based on Figure 4, what kind of population growth model would you use to represent the Serengeti wildebeest population? Why? The wildebeest population will follow the shape of the logistic growth model. The population size increases, then it grows slow and the supports the constant value of the carrying capacity. 16. Was the wildebeest population at the carrying capacity in 1968? Why or why not? No, the population in 1968 was not in the carrying capacity because it grows until around 1980. 17. Calculate the size of the wildebeest population in the year 1968, using the logistic model simulator with the following settings: K = 1,245,000 wildebeest, r = 0.2717 per year, and N0 = 80,000 wildebeest in the year 1958. 634,497 wildebeest.

18. Imagine that the maximum per capita growth rate (r) for the wildebeest population increased to 0.4 per year in 1958. a. Suggest a specific reason that r could increase for a population. It could be that there were multiple factors that that can affect r, that can be the birth, death, immigration, and emigration. b. Recalculate the population size in 1968 using the new r. You can use the same values for the other settings as in Question 17. 982,852 wildebeest. c. Sketch or describe how the wildebeest population curve in Figure 4 might change as a result of the new r. The population would be growing faster and it would reach their capacity more quickly. 19. Imagine that the carrying capacity (K) for the wildebeest population decreased to 1,000,000 wildebeest in 1958. a. Suggest a specific reason that K could decrease for a population. The habitat could become smaller and part would be becoming into cities and farmlands, and the less it rains it would decrease the places they eat. b. Recalculate the population size in 1968 using the new K. You can use the same values for the other settings as in Question 17. 568,235 wildebeest c. Sketch or describe how the wildebeest population curve in Figure 4 might change as a result of the new K. The population would be growing more slowly and it would stop growing onto it reach the smaller carrying capacity. 20. Look at the size of the zebra population, which is shown as triangles in Figure 4, before and after the

rinderpest vaccination campaign. a. What patterns or trends do you observe in the zebra population? The size of the zebra does not change, during and after the rinderpest campaign. b. Based on your answer above, what effect does rinderpest have on zebras? It does not affect the zebras at all, because the rinderpest had no effect on them. 21. Based on Figure 4, did the zebra population growth rate (dN/dt) differ in the years 1985 and 2003? Why or why not? (Hint: dN/dt is equal to the slope of the curve showing population size, N, over time, t.) The zebra growth rate was nearly 0 in both years 1985 and 2003 and it didn’t differ in these years. 22. Imagine that there is a large wildfire in the Serengeti in 2010. a. How might the zebra and wildebeest populations change right after the wildfire? The fire would kill both of them, the population would decrease their size. b. How large do you think the zebra and wildebeest populations would be 50 years after the wildfire? Explain your answer, or what else you would want to know before making a prediction. If the fire, have not damaged several, the population could return to the same carrying capacity as before the fire, but this will be taking the time....