Bio 2550 Population Viability Analysis Lab

Bio 2550 Population Viability Analysis Lab

Due noon October 23, 2012

Note: in order to complete this assignment you will answer questions based on calculations you do on the laptops provided. You will find it useful to copy the files you create in this exercise onto your Flash Drive.

Populations of organisms that are in danger of extinction are often monitored for the purposes of determining their viability, i.e., whether they will persist for the foreseeable future. Population Viability Analysis (PVA) is one of the most common analytical tools used to examine population persistence. In this lab, we will examine two PVA methods. The nonintegrated method uses the number of individuals in the population. The advantage of the nonintegrated method is that data collection is relatively easy. The disadvantage is that the nonintegrated method assumes that all individuals of the population have an equal chance of survival, growth and reproduction. This assumption is rarely true. Recall, for example, that for species with a Type III survivorship curve, mortality is very high in young individuals and very low in older ones.

In order to avoid the problems of the nonintegrated method, and hopefully provide a more accurate estimate of population viability, conservation biologists often use the integrated method. The integrated method divides the population into classes based on age, size or stage and follows the fate of individual organisms. This information is used to determine the probability that an individual that is a member of a class in year 1 will remain in that class, move into another class or die in the following year. The major drawbacks of this method are the time consuming method of gathering data and the more complicated method of analysis.

To compare the two PVA methods, we will use data from Savanna Blazing Star (Liatris scariosa var. nieuwlandii), a prairie and savanna species listed as threatened with extinction in Illinois. There are fewer than 30 populations in Illinois, largely due to habitat destruction. The life cycle of Savanna Blazing Star is depicted in Figure 1.

The largest population of Savanna Blazing Star occurs in Hickory Creek Barrens Nature Preserve in Will County, IL. Individuals of Savanna Blazing Star were monitored from 1995 to 2000 in plots that differed in the amount of cover of the grass, Big Bluestem (Andropogon gerardia). A portion of the demographic data is reproduced in Table 1 on a separate handout. Each pair of students will receive a different portion of the data.

Note: for some of the analyses, we will use an Excel add-in called PopTools. This software can be downloaded for free from the website listed in the References, but is only available for use on computers operating Windows. On-line help is available in the “Demos” menu option.

Submit your answers to the questions below through Moodle using the PVAQuestions.doc file.

Nonintegrated Method

First we will do some calculations using the total population numbers.

1.  [1 point] Indicate below the following information from Table 1.

Bluestem cover class:

State year:

Fate year:

2.  [1] State year total (Ts):

(Add all the plants in the “Total” row at the bottom of Table 1. Notice that the total for the “flowering” column does not include the value in the first row. This is intentional. Why do you suppose that the first row is not included in the column totals?)

3.  [1] Fate year total (Tf):

(Add all the plants in the “Total” column, except the “Dead” row.

4.  [1] Calculate estimated annual finite growth rate (l, lambda):

(Divide fate year total by state year total: l = Tf/ Ts)

5.  [2] Does the annual finite growth rate indicate population growth? How do you know?

6.  [1] Calculate estimated intrinsic growth rate (r = lnl):

7.  [2] Does the intrinsic growth rate indicate population growth? How do you know?

8.  [1] Calculate the estimated population size for the next year (Tf * l):

Integrated Method

Next, we will calculate the same values using a matrix of stage transitions.

9.  [5] Using the values in Table 1, calculate the transition frequencies for each stage. Transition frequencies are calculated by column. For each column, divide each value by the total and place these values, to three decimal places, in Table 2, below. Now add the frequencies for each column, excluding the first row. The total for each column should equal 1.0.

Table 2. Transition frequencies for Savanna Blazing Star stages.

state
fate / seedling / juvenile / flowering / vegetative / dormant
seedling
juvenile
flowering
vegetative
dormant
dead
Total

10.  [2] Calculate the annual finite growth rate (l):

(This is more complicated compared to the nonintegrated method, so we’ll use some software called PopTools. Copy the transition matrix above into an Excel spreadsheet. Be sure to fill all the zeros in your matrix.

. Select the five columns of values and the first five rows of values. Go to the PopTools menu and select “Matrix Tools” and then “Finite Rate of Increase”. In the dialog box, indicate a destination. Click the “Go” button. The value entered into the destination cell is l)

11.  [2] How does the value of l using the integrated method compare to the value of l using the nonintegrated method?

12.  [5] Calculate the estimated population size for the next year. (See example on p 36, Morris et al., 1999 - this document is available on Blackboard). We do this by multiplying our matrix of transition frequencies by a vector of the number of individuals in each stage class for the fate year. Instead of doing this by hand, however, we will create formulas in Excel to perform these calculations for us. First, in a column to the right of your transition matrix, enter the number of individuals in the fate year for each class. You should have something that looks like this, where the letters represent the transition frequencies:

matrix / vector
a / b / c / d / e / 100
f / g / h / i / j / 50
k / l / m / n / o / 20
p / q / r / s / t / 10
u / v / w / x / y / 5

Now, create a new column directly to the right of your fate year vector. In this column, you will write formulas to calculate the new numbers of individuals in each class for the following year. Start by selecting the first cell of the column and type “=” and “(“. These will show up in the formula bar at the top of the page. Then click on the upper left cell (the one I have labeled “a”), followed by “*”. Then click on the first cell of the vector column (the one I have labeled “100”) followed by the “)” and “+”. Continue adding to the formula (see below) until each value in the fate vector is multiplied by a value in the first row of the matrix (i.e., the vector value in 1st row is multiplied by the matrix value in the 1st row and 1st column, the vector value in the 2nd row is multiplied by the matrix value in the 1st row and 2nd column, etc.). Once you’ve completed the formula, hit the “enter” or “return” key and the formula will be replaced with a value. This value is estimated number of individuals in the first stage class next year.

matrix / vector / next year’s population size
a / b / c / d / e / 100 / = (a*100)+(b*50)+(c*20)+(d*10)+(e*5)
f / g / h / i / j / 50 / = (f*100)+(g*50)+(h*20)+(i*10)+(j*5)
k / l / m / n / o / 20
p / q / r / s / t / 10
u / v / w / x / y / 5

Write the formulas into the remaining cells following the same procedure until you have values for all five classes. Now, sum the column to calculate the total population size for next year, and write the answer here.

13.  [2] How does the estimated population size for the next year using the integrated method compare to the value using the nonintegrated method?

In order to project the population for 10 generations, and to see whether your calculations were correct, we will use PopTools again. Select your matrix, then “PopTools”, “Matrix Tools”, then “Matrix Projection”. The dialog box asks you to select your vector and an output cell. Do that and Click “Go”. You’ll see a table with “Time” from 0 to 10 and the values for each stage. Let’s graph this. Select the 5 columns labeled “Stage” and click on the Chart Wizard. Select the Line chart type and click “Next”. In the next dialogue box click on the “Series” tab. Click on the “Category (x) axis labels” at the bottom of the dialogue box and select the values 0 through 10 under “Time”. Click Next. Fill the Chart title and axis names. Click Finish.

14.  [3] Look at time 0. The numbers for each stage should be the same as your vector values. Now look at time 1. The numbers for each stage should be the same as the values you calculated for next year’s population size. If not, look at your formulas and determine where you may have made an error. Describe the trend in stage numbers over the ten projection intervals.

Next we’ll use PopTools to do a more extended matrix analysis. Select the matrix again. Choose “PopTools”, “Matrix Tools”, and “Basic Analysis”. In the dialogue box, select “Color Output” and the output range. The output included several regions indicated by different colors. The pink region in the upper left is called “eigenvalues” and the value in the upper left is l. We will ignore the remaining values in the pink region.

15.  [2] The yellow region is labeled “Age/stage structure” and is the stable age or stage structure, meaning the proportions of individuals we would expect for each stage, even as total population density grows or declines. Look at your graph. Are the proportions of your stages stable (i.e., do they remain the same from year to year)?

16.  [2] The darker blue region is labeled “Reproductive Value”. This is the relative contribution that an individual in a particular class is expected to make towards future population growth. Do all stages have equal reproductive value? Why not?

17.  [2] In the light blue region, we will look only at “r”. How does the value of r using the integrated method compare to the value of r using the nonintegrated method?

In order to answer the following questions, everybody’s integrated and nonintegrated values for population size and l will be compiled and graphed.

18.  [3] How do the values for population size and l (integrated and nonintegrated) compare from year to year?

19.  [3] Is an observation in a single year sufficient to evaluate the persistence of a population? Why/why not?

20.  [3] How do the values for population size and l (integrated and nonintegrated) compare between Big Bluestem cover classes?

21.  [2] Based on this data, will a population of Savanna Blazing Star growing with 0% bluestem cover persist? How do you know?

22.  [2] Will a population of Savanna Blazing Star growing with 1-50 % bluestem cover persist? How do you know?

23.  [2] Will a population of Savanna Blazing Star growing with 51-100% bluestem cover persist? How do you know?

References

William Morris, Daniel Doak, Martha Groom, Peter Kareiva, John Fieberg, Leah Gerber, Peter Murphy, and Diane Thomson. 1999. A Practical Handbook for Population Viability Analysis. The Nature Conservancy.

Hood, Greg. PopTools. http://www.cse.csiro.au/poptools/ CSIRO, Canberra, Australia.

Figure 1. Generalized stage-based population dynamics model for Savanna Blazing Star (Liatris scariosa var. nieuwlandii) presented as a graphical network. Circles represent stages, arrows represent possible transitions between stages, arrow thickness increases with transition probability.

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