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# Population Dynamics

Population Dynamics Lab Table Number: Group Member Names: (First, Last) Section: Date: Group Member PIDs: Introduction In Ecology, a population is defined as a group of individuals of the same species found in a defined area. Some examples of populations are all the humans in the entire globe, deer in the state of Ohio, and E. Coli in a human. Population Dynamics studies how a population’s size and/or density changes over time. The study of population dynamics helps scientists identify, monitor, and manage endangered species. Environmental resource managers also use population dynamics to ensure stable food supplies for humans. Many populations are challenging to study because they occupy a very large space, or the organism is hard to find. Therefore, population ecologists have to find other ways to study populations. A common method for studying populations is by using mathematical models. In ecology, models are simplified mathematical representations of phenomena used for research purposes. As it happens in other fields (e.g., weather forecast), population models allow ecologists to predict changes in population size based on multiple factors (biotic and abiotic). In theory, four primary factors can dictate population size, measured as the number of individuals at a given time. Those factors are immigration (individuals entering the population), emigration (individuals leaving the population), number of newborns, and number of deaths. The simplest population model in ecology assumes a close population with NO immigration or emigration. Therefore, only the number of newborns and deaths could change the population size. If the number of newborns is greater than the number of death, the population size (the number of individuals) increases (positive growth). In contrast, if more individuals die than the number of newborns, the population will decline (negative growth). The second big assumption is that there are no limiting factors. Thus, every potential resource used by the species (space, food, water, mates) is enough, and thus there is no competition. Under these conditions, the model predicts exponential growth (Exponential model). Several species, such as mosquitos, rabbits, and mice, could display this exponential growth. If individuals in a population compete for one or more resources, as the number of individuals increases, the resource(s), they compete for become less available to the point where the population reaches the maximum number of individuals as a result of competition. In that case, the mathematical models display an “s” shape curve. We call this model the Logistic Growth Model. You can see in the figures that the first portion of the model (up to the inflection point) is identical to the exponential model. Still, it changes after that point until the population reaches its maximum carrying capacity (maximum number of individuals given the amount of resources available to the population). Other factors, such as stochastic events (unexpected environmental changes such as hurricanes) and predation, can impact population dynamics. We will be doing simulation modeling, a method that allows us to predict the future state of a population. We will use Microsoft Excel and NetLogo to model populations in today’s lab. QUESTIONS 1. What is Ecology? The study of how organisms alter the environment for their own benefit or benefit or other living organisms. 2. What is population dynamics? 3. Give one example of a population that was not given in the introduction. 4. Give one example of a model that you learned about in another class. Task 1: Exponential Growth One of the earliest and simplest populations dynamics models is the exponential growth model. Very few real-world populations exhibit true exponential growth, but some do, like bacterial populations. For our exercise today, we will use E. Coli as our example. Exponential Growth Equation: 𝑁𝑡+1 = 𝑁𝑡 + 𝑟𝑁𝑡 N – population size t – time r – intrinsic growth rate. If r > 0, population is growing; if r < 0 population is declining, if r = 0 population is not changing. Number of births minus the number of deaths per generation time E. coli is a gram-negative, rod-shaped bacterium that is normally found in the intestines aiding during digestion. Through our study of symbiosis, we know this is a mutualistic symbiosis. However, E. coli is still one of the most common pathogens usually encountered. The most common way of getting infected is by eating contaminated food or drinking contaminated water. Symptoms include vomiting, diarrhea, stomach pain, and fever. Today we will study the bacterial population of an individual that became infected by swimming in a contaminated river. PROCEDURE PER STUDENT 1. Download and open the population dynamics Excel template found on Canvas. 2. Once you have the template open in excel, make sure the “Task 1 Exponential Growth” Tab is open. 3. Since our patient was only infected by one E. coli to start with. Enter a 1 into cell B2 as the initial population size. 4. Enter the following formula into cell B3, “= B2 + B2 * $E$2”. What does the = do in Excel formulas? What does the $ do in Excel formulas? 5. Enter 0.1 into cell E2. Cell E2 holds the value for r, the intrinsic growth rate. 6. Click the small box in the right corner of cell B3 and drag down the formula until cell B62. 7. Create a graph. a. Go to the “insert” tab at the top of Excel b. Find and press the button to add a “Scatter plot with smooth lines.” c. Click on your graph so that you get “Chart Design” tab at the top of your screen. d. Press the “Select Data” button, then press the “Add” button in the new window. e. In the series name box, type in “Ecoli” f. Click the Series X values box, then use your mouse to highlight all the Data in your “Time” column g. Click the Series Y values box, then use your mouse to highlight all the Data in your “N” column. After this is finished, press the “OK” button h. Change the title of your graph to “Exponential Population Growth” i. In the “Chart Design” tab, press the “Add Chart Element” button and add X and Y axis titles. Label the x-axis as “Time”, and label the y-axis as “Population Size (N).” QUESTIONS 1. Draw the graph you made in Excel. Make sure to include Graph and Axis Titles. 2. Fill in the following table using excel using data from question 1. Time step Population Size (N) 0 1 20 6.7275 60 276.801 3. Given that intrinsic growth rate (r) is the difference between the number of newborns and the number of deaths, how would the shape of the exponential growth model change as the number of newborns increases (holding constant the number of deaths)? Represent your initial curve (the one you use above) and your predicted curve. Make sure you label the axis and include the units. 4. Fill in the following table using excel using data from question 3. Time step Population Size (N) 0 20 60 5. How is your curve in question 3 different from your curve in question 1? 6. Let’s investigate how changing the starting population size influences your population growth curve. Pick an N0 between 50-100 and write it below. 55 7. Enter the new N0 into your excel model and draw your new graph. Make sure to include axis-titles, tick marks, etc. 8. Fill in the following table using excel using data from question 3. Time step Population Size (N) 0 20 60 9. How is your curve in question 7 different from your curve in question 1? 10. Several species of insects can display Exponential Growth, indicative of low to no competition for resources. If limiting factors are not the population growth regulators, what factors can control these populations? Give an example. Excessive hunting, genetic drift: Ex: Bengal Tigers Task 2: Logistic Growth In the example above, we created our model based on the assumption that resources are infinite, which tends not to exist in the real world. In actual population, exponential growth may happen at first when the population is small, and resources are plentiful. But when the number of individuals increases in the population and the amount of resources are used up, the growth rate of the population will start to plateau. The curve in a logistic growth model resembles an “Sshaped.” The population size at which the curve begins to plateau represents the maximum population size in a given environment, which is called the maximum carrying capacity, or K. Going back to our previous example with the subject infected with E. coli, we are going to simulate how the bacterial population will change with a given carrying capacity. After being infected, the invasive population will rise exponentially, but more resources need to be used up as the population increases. As a result, more invasive bacteria will need to compete with the host’s bacteria for survival, and the population of invasive bacteria will reach a maximum number. There could be a case where the population could exhaust its resources. The Malthusian Dilemma embodies such a scenario. Thomas Robert Malthus proposed this. Such a concept is that population growth is exponential and resource growth is linear. In a Malthusian model, the point of intersection between population growth and resource growth is called the Malthusian catastrophe, which leads to a struggle for survival and an increase in mortality rate. Logistic Growth Equation: 𝑁𝑡 𝑁𝑡+1 = 𝑁𝑡 + 𝑟𝑁𝑡 (1 − ) 𝐾 N – population size t – time r – intrinsic growth rate. If r > 0, population is growing; if r < 0 population is declining, if r = 0 population is not changing. K – carrying capacity PROCEDURE PER STUDENT 1. Download and open the population dynamics Excel template found on Canvas. 2. Once you have the template open in excel, make sure the “Task 2 Logistic Growth” Tab is open. 3. Enter a 1 into cell B2 as the initial population size. 4. Enter the following formula into cell B3, “= B2 + B2 * $E$2 * (1 – (B2/$F$2))”. 5. Enter 0.2 into cell E2. Cell E2 holds the value for r, the intrinsic growth rate. 6. Enter 25 into cell F2. Cell F2 holds the value for K, the carrying capacity. 7. Click the small box in the right corner of cell B3 and drag down the formula until cell B62. 8. Create a graph. a. Go to the “insert” tab at the top of Excel b. Find and press the button to add a “Scatter plot with smooth lines.” c. Click on your graph so that you get “Chart Design” tab at the top of your screen. d. Press the “Select Data” button, then press the “Add” button in the new window. e. In the series name box, type in “E.coli” f. Click the Series X values box, then use your mouse to highlight all the Data in your “Time” column g. Click the Series Y values box, then use your mouse to highlight all the Data in your “N” column. After this is finished, press the “OK” button h. Change the title of your graph to “Logistic Population Growth” i. In the “Chart Design” tab, press the “Add Chart Element” button and add X and Y axis titles. Label the x-axis as “Time”, label the y-axis as “Population Size (N)” QUESTIONS 1. Draw the graph you made in Excel. Make sure to include Graph and Axis Titles. 2. Use your graph in excel to determine at what time does the inflection point occur? (At what point in time does the graph start to plateau?) 3. What kind of growth happens before the inflection point? How can you tell? (Exponential or logistic) 4. What kind of growth happens after the inflection point? How can you tell? 5. What would happen to the bacterial population if the r parameter was less than 1? Explain your reasoning 6. What does it mean for a population to r-selected? K-selected? 7. Imagine you own a five acres property where you keep a small sheep population (n=10). Over time (15 years), the population increased (fast initially) up to 50 individuals. The population has fluctuated around 50 individuals for the last five years. The numbers show a logistic growth model. A) Draw the potential growth model of this population. B) What options can you propose if you were to increase the size of this population? (Use excel to answer this question) Task 3: Mark-Recapture Estimating population size is a challenging task for ecologists. In terrestrial or aquatic ecosystems, cryptic behaviors, highly mobile species, and ecosystem physical-structural complexity (e.x., rain forests and coral reefs) can make population surveys quite tricky. One technique to estimate population size is mark-recapture, where ecologists capture, mark, and release a small number of individuals. On a later day (e.x., a week later), another capture takes place where the number of individuals marked, and unmarked are recorded. Usually, in a small population, you are more likely to recapture marked individuals, whereas, in a large population, you are less likely. Populations of green sea turtles in the Caribbean, sharks, and large birds such as the California Condor are typically surveyed using mark-recapture techniques. Mark Recapture Equation: 𝑁= 𝑀𝐶 𝑅 N = Estimated Population Size M = Number of Individuals Marked C = Number of Individuals caught in the Census Period R = Number of Individuals caught in the Census Period that are Marked PROCEDURE PER STUDENT 1. Open up the following link to the Mark-Recapture simulation. This link should also be found on Canvas. http://virtualbiologylab.org/NetWebHTML_FilesJan2016/PopulationEstimationModel.ht ml 2. Press the “Setup” button, then press the “Go” button. You should see the tadpoles start to swim around in the simulation model. 3. While the simulation is running, press the “Dip Net” button. This will move Tadpoles into your sample bucket. 4. While Tadpoles are in your sample bucket, press the “Mark button” until all the tadpoles in the backet are pigmented red. 5. Move Tadpoles from the sample bucket into the holding pen by pressing “Hold” button. 6. Repeat steps 3-5 two more times. 7. Write down the number found in the Total Marked box under the model simulation. 8. Release Tadpoles in Holding Pen back into the Pond by pressing the “Empty Pen” Button. 9. Press the “Dip Net” button three times. Write down the total number of individuals in the bucked and the number of marked individuals in the bucket. 10. Press “Go” to stop the model. Press “Setup” again to reset setting to default. 11. Use the Mark-Recapture Equation to estimate the population size of tadpoles inside the pond. QUESTIONS 1. Fill out the data below using the NetLogo Model Parameter Count M C R 2. Use your data table above to calculate the number of tadpoles in the pond. (Show your work!) 3. Repeat the experiment again but change the size of your dipnet to large. Record your data again in the table below? (Remember to stop the model and press “setup” after changing your setting before recording data) 4. Use your data table above to calculate the number of tadpoles in the pond while sampling with the large net. (Show your work!) 5. Is your population estimate in question 4 (big net) similar to your estimate in question 2 (medium net)? If not, why do you think the estimates are different? 6. Scenario: You gather a sample of 20 horseshoe crabs, and mark release them. If you return the following week and collect 30 horseshow crabs and six possess marks, how large would you estimate the population to be? 7. What if instead of 30, when you return you’ve caught 50 horseshow crabs and 10 are marked? Does this change your population size estimate? 8. Discuss with your peers. What factor(s) could affect the Mark-Recapture method? Task 4: Model your own system GROUP INVESTIGATION As a table, work together to model your own population. Follow the steps below! Pick a real-world population you want to model? Is your population r-selected or k-selected? Should your population show exponential or logistic growth? What is the starting population size of your population? (try to be realistic) Use google to estimate the population growth rate of your population. If you cannot find one, estimate it as best you can. Report it below. If your population has a carrying capacity, try to google it or estimate it below. Use excel to model the population you group came up with. (model 200 time steps, you will have to adjust your graph and data sheet) QUESTIONS 1. Enter data from your modeled population in the table below. Time step Population Size (N) 0 50 100 150 200 2. Draw the graph of your modeled population. Remember to include axis titles, tick mark, etc. 3. Do you believe your model accurately reflects the real-world population? How could you improve your model. 4. What are some real-world factors that influence your population? 5. Models are an important research tool in many sciences including Biology. However, they are not always completely accurate to the real-world system. Why do you think they are still used in scientific research? Sources Gotelli. A Primer of Ecology 4th edition. 2008. Sinauer Associates, Inc. Time N 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 r 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 place initial parameters in green cells place initial formulas / equations in yellow cells Exponential Growth: 𝑁𝑡+1 = 𝑁𝑡 + 𝑟𝑁 𝑟𝑁𝑡 Time N 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 r K Logistic Growth: 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 tic Growth: 𝑁𝑡+1 = 𝑁𝑡 + 𝑟𝑁𝑡 (1 − 𝑁𝑡 ) 𝐾 Time Step N 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 r K 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198