Pennsylvania State University
BIOL 463 Homework 1 -- Demography and Population GrowthHow to complete this assignment. This assignment is divided into two parts. There are 9 questions in Part 1, 2 questions in Part 2, and one extra credit question. There is an associated Excel spreadsheet file that you will be directed to use to complete the assignment. There are also two journal articles that you may read and consult to complete the assignment. Please type your answers in a separate file (that contains your answers only – any text format will do) and submit them to me in the Homework 1 dropbox on the course ANGEL website. In Part 1, you will also be asked to save and submit a filled-in copy of the Excel spreadsheet. Please name both the answer file and the Excel spreadsheet with your PSU ID – e.g. ABC1234.doc and ABC1234.xls.Please use complete sentences for all answers.This assignment must be uploaded to ANGEL by 5PM on February 5.Part 1. Life Tables and Reproductive ValueElk are a dominant herbivore in the Greater Yellowstone Ecosystem, numbering >120,000 animals living in the area that includes Yellowstone National Park itself and the contiguous national and state forests in Montana, Wyoming and Idaho. Prior to 1995 the major source of mortality in the elk population was harvest due to human hunting and mortality due severe winter conditions in the area. In 1995, 31 wolves were reintroduced into Yellowstone National Park. The wolf population quickly grew to 132 animals by 2001, prompting concern among tourism and hunting groups about the impact of wolf predation on the growth and maintenance of the elk population. In 2006, Wright et al. published a life table analysis of the elk population with the goal of assessing the impact of wolf predation relative to other sources of mortality in the elk population. The results of that life table analysis are presented in the excel spreadsheet associated with this assignment (sheet 1). In the first part of this assignment, review both the analysis in the excel spreadsheet and the Wright et al (2006) paper itself, and answer the associated questions. In the sections below, I provide calculations for two additional measures: life expectancy and reproductive value. Reproductive value: A useful statistic that can be calculated from the life table is the reproductive value (Fisher 1930). The reproductive value is the relative number of offspring that remain to be born to individuals of a given age. Though it is intuitive to assume that the reproductive value is maximized in newborns (you have your whole life to reproduce), a newborn’s reproductive value must be discounted against the fact that a newborn may not achieve its full reproductive lifespan. Thus, an individual that has made it past the initial developmental stages may have higher reproductive value because it is guaranteed to have survived long enough to reproduce. Let v(x) be the reproductive value of an individual of age x, then the reproductive value is defined as:vx=# of offspring produced by age x or oldernumber of individuals of age xThe numerator accounts for all the offspring left to be produced in ones lifetime, and the denominator discounts for the probability of achieving the rest of ones lifetime. Using the notation of the life table, we can write v(x) as vx=erxlxy=x+1ke-rylxmxThe Elk Life Table: On sheet 1 of the Homework 1 Excel file (the tab called “Life Table”) you will find the elk life table from the Wright et al paper. The cells in grey are the sampled values for elk survival and fecundity from that paper. The cells in blue are blank, and you will be asked to fill these values in. The cells in green contain the formulas to calculate the reproductive value (above) – ignore the initial values in these cells, they will fill in with the correct values as you fill in the life table. Below the table are cells in yellow where you will be asked to fill in the summary values for the life table. (Note: I realize that all these values exist in the paper and you could simply fill them in without performing any calculations –I changed a few values so the calculated values won’t match the paper exactly. Since there’s a reasonable chance you’ll be asked to do these kind of calculations on an exam, it might be worth proving to yourself that you can do it for real.)Complete the following questions/task that pertain to sheet 1 of the excel file. Note: you may calculate the necessary values by hand, with a calculator, or using function capabilities in Excel. For the questions below, present give all answers to 2 decimal places unless otherwise specified. Calculate and fill in the values for the blue columns (lxmx and xlxmx).Using those values, calculate the basic reproductive value, R0 and fill it in the corresponding yellow box at the bottom of the table.Calculate the generation time, Tc, and fill it in the corresponding yellow box at the bottom of the table.Calculate the intrinsic rate of increase, r, and fill it in the corresponding yellow box at the bottom of the table (provide this answer to 4 decimal places.).When you have completed questions 1-4, save a copy of the Excel worksheet. Rename the file with your PSU ID (i.e. ABC1234) and submit the file (make sure it has the values for 1-4 filled in!) to the Homework 1 Dropbox.As you complete questions 1-4, the values in the green columns should start changing (because their functions depend on the values you are calculating).On average, how many female offspring is one cow elk expected to have in the course of her lifetime.Wright et al. constructed this life table using demographic rates and population counts from animals harvested by hunters between (1996-2001); i.e. this is a static rather than a cohort life table. Describe how using hunter-killed elk may bias the estimate of the age distribution of the elk population.What age has the highest reproductive value? What is it?Wright et al. concluded that wolf predation should have a smaller impact on elk population growth than mortality due to hunters based on an analysis of the age-specific reproductive value. Describe the quantitative basis for this conclusion.Part 2. Environmental Variability and Population GrowthThe calculation of basic reproductive rate and the intrinsic rate of growth provide a measure of the expected rate of population growth. In practice, that growth rate is likely to fluctuate from year to year as a consequence of environmental variability (e.g. El Nino years, droughts, hurricanes, severe winters). Thus the growth rate of any particular population may best be described as a distribution with a mean equal to the life table value, and some variance determined by how much the local environmental conditions fluctuate. Brian Dennis (1991) and colleagues considered the impact of this environmental fluctuation on the growth populations of endangered species – and in particular considered the probability that a population with, on average, positive population growth rate (i.e. as might be calculated from a life table), might go extinct due to random environmental fluctuations. Using the associated excel spreadsheet, I would like you to explore the impact of environmental variability on population growth and extinction probability for endangered populations. Deterministic and Stochastic Growth models: The exponential growth model that we discussed in class is deterministic (i.e. there is no random fluctuation in values) – as such the population at any one time can be calculated directly from the initial value (N0). In the real world, population will be higher or lower than expected following particularly good or bad year, so the value of the population size at time t depends on the value in time t-1, not just on the initial conditions. The second sheet of the Excel file (“Population Projections”) calculates and plots the projected exponential growth for a deterministic model, and one where the annual growth rate fluctuates from year to year. Below I will ask you to manipulate this sheet and summarize the results to investigate how environmental variability may affect the growth of a population.Orienting to the Excel sheet:On the left of the sheet are 5 number that correspond to the important parameters of the population model:Initial population size is the starting number of animalsBasic reproductive rate, R0.The generation time, Tc.The intrinsic growth rate, r.The environmental variability in the growth rate, V. This indicates how much the intrinsic growth rate, r, varies from year to year. Technically, the growth rate each year is drawn from a normal distribution with variance equal to V. Basically, the bigger this number is, the bigger the chance of really good or really bad years. Note, the chance of good years are equally as likely as bad years. To the right of that are 3 columns of numbers:year is the time in years.“Deterministic Population size” is the projected population size each year for the deterministic exponential growth model given the parameters and starting population size (at left).“Mean of Stochastic Runs” gives the average projected population size each year for 100 randomly generated population trajectories given the parameters and starting population size (at left).If there are enough bad years, the population size might drop below 1 individual, thus going extinct. The “Proportion of Stochastic runs still alive” indicates the proportion of those random simulations where the population has stayed above 1 individual (i.e. the probability of the population not going extinct) for each year.To the right of that is a figure that shows the projected population models over time. The thick solid red line indicates the deterministic model. All the other lines are the individual random simulations (these are all calculated using the remaining sheets). Notice that the lines that drop to 0 and stop are the simulations in which this population went extinct. Note: the y-axis is on the log scale, so exponential growth looks linear.NOTE: Every time you calculate something new on the sheet, it will generate new random simulations – so the two right most columns of numbers and the figure will change. This is supposed to happen! You can make the sheet calculate a new set of stochastic simulations by pressing F9 (or function+F9 depending on your keyboard set up) – you’ll need to do this to complete the assignment.Using this Excel sheet, I would like you to do a simulation experiment to quantify how extinction probability responds to 3 variables (initial population size, generation time, and environmental variability) when the basic reproductive rate is held constant. Please fill in the values of N0, Tc, and V for each of the 8 scenarios in Table 1 and fill in the proportion of the simulations in which the population survived for 25 years. Using F9, recalculate this value for each set of parameters 3 times and fill it in the table. NOTE: keep R0 constant at 3; the intrinsic rate of increase with calculate automatically. Table 1. Proportion of populations surviving for different parameter values.InitialGenerationEnvironmentalIterationPopulationLengthVariability123125.22210.2325.44210.45105.261010.27105.481010.4Questions: Fill in the values for the proportion of the simulations in which the population survived for 25 years for three iterations of each of the 8 parameter combinations in Table 1. Please copy and paste the table, into the file with your responses. Describe the relationship between the 3 variables tested in the simulation experiment above (Table 1) and the extinction probability. Describe the direction of the relationship (i.e. extinction probability goes up as X goes down) and provide a justification for why this relationship makes sense.Initial population size:Generation time.Environmental variability.Extra Credit:Consider the re-introduction of wolves from section 1 of this homework. Following re-introduction, the Yellowstone wolf population was very successful and clearly did not go extinct – in part due to the successful efforts of the wildlife managers overseeing the reintroduction. Referring to the patterns in from this simulation experiment, suggest 2 specific steps that wildlife managers could have taken to increase the probability of successful reintroduction of wolves to Yellowstone. ................
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