Lab 12: Population Viability Analysis- April 12, 2004 DUE: April 19 2004 at the beginning of lab Procedures: A. Complete the workbook exercise (exercise 28). This is a brief exercise and provides needed background for the structured model you will build next. Skip the final step of extending your simulation to 100 years. Answer the following questions (5 points): 1) a. How does the initial population size of our model influence the probability of extinction? Alter cell B4 as listed in the table below, and press F9 10 times. Record the probability of extinction for each trial. Find the average probability of extinction for each population size. Initial Population Size Trial 10 25 50 100 500 1 2 3 4 5 6 7 8 9 10 AVERAGE b. Repeat the process above using a standard deviation of 0.4 (cell D4). Does decreasing the variation have any effect on your probabilities of extinction? Interpret your results. B. Complete the exercise on spotted owls included below. What to hand in (15 points): 1) For Part 1 (your spreadsheet titled 'Spotted Owl Management Options I'), provide a few sentences explaining which scenario you believe is best for maintaining the population of spotted owls, and why. Provide your graph of the probability of extinction for the 10 trials of each scenario. 2). For Part 2 (your spreadsheet title 'Spotted Owl Management Options II') again provide a few sentences explaining which scenario you chose and why. Also compare your results in part 2 to your results in part 1. 3) Provide your table of extinction risk for the three management scenarios for your original model, your model with doubled variation, and your model with halved variation. Provide a few sentences addressing the effects of variation on the outcome of your PVA.
Population Viability Analysis II. The Case of the Spotted Owl (Strix occidentalis caurina) Population viability analysis is most defensibly used for either determining which components of a population model are most critical and therefore need accurate estimation, or for comparing possible management scenario outcomes as a way of trying to assess possibilities before any management plan is actually implemented. In this part of the lab, you will create a simple PVA model for spotted owls, and rank several different management options. Set up: Create a new page in the workbook already started with the Donovan and Welden exercise. Title the page Spotted Owl Management Options I in the upper right corner. You will calculate the asymptotic rate of increase for a matrix model, then use that asymptotic lambda value as the basis for the stochastic modeling. In this exercise, we will leave the actual demographic rates as point estimates. Another way of modeling stochastic variability is to incorporate the variability into the demographic estimates themselves. You should consider doing this on your own, some pointers will be provided at the end of this handout. The projection matrix we ll use for the spotted owls is based on data collected by Noon and Biles (1990: Journal of Wildlife Management 54:18-27). In cells b4-d6, type in the following matrix: 0 0 0.226 0.11 0 0 0 0.71 0.942 In F3, type Start Vector, and fill in 70, 50, and 25 in cells F4-F6, respectively. Set up the headers Mean lambda, Std lambda and Extinction in cells H3-J3. In H4, type =$F$37, in I4, type 0.3, and in J4, type =1-AB66. In A9, type Calculate the Asymptotic Rate of Increase, H9 Trial, and J9 Lambda Value. In cell A11, type time, B11 type N0, C11 type N1 in D11 type N, in E11, type Total, and F11, type lambda In the time column, fill in a series of 0-25 (A12 A37). Cells B12-F37 are going to be filled with the calculations of the total numbers of owls and the asymptotic rate of increase; we ve done this before. In B12, type =$F$4, C12 =$F$5, and D12=$F$6. In E12, type =sum(b12:d12). In B13, type =B12*$B$4 + C12*$C$4 + D12*$D$4 -- the number of new owlets in year 1. In C13, type =B12*$B$5 + C12*$C$5 + D12*$D$5 -- the number of subadults in year 1. In D13, type =B12*$B$6 + C12*$C$6 + D12*$D$6 the number of adults in year 1. In F13, type =E13/E12. Copy E12 formula into E13. Copy the formulae in B13 to F13 down to B37 to F37. Notice that the population growth rate hits the asymptotic rate of increase well before time step 25.
Now you will create a series of stochastic population growth estimates. In H10, enter 1. In H11, enter =H10+1. Drag this down to H34 for 25 time steps. In I10, enter =NORMINV(RAND(),$H$4,$I$4). Copy this across to AG10, and down to AG34 to give you a 25 x 25 matrix. Now you will calculate actual numbers of owls through time. In cell B39, type Year, and in A40, type trial. B40-AA40 will be the number series 0-25. In cells A41-A65 enter the number series 1-25. In AB39, type Persist? and in AB40, type 1 = yes, 0 = no. In B41, type =$E$12. Copy this down the column to B65. In C41, type =ROUND(IF(B41*I10<1,0,B41*I10),0). This equation multiplies the starting population size by the stochastic population growth rate. Copy the equation across to AA41. Check your formulas, then go ahead and copy your formulas in C41-AA41 down to C65-AA65. In cell AB41, type =IF(AA41>0,1,0). This equation gives a value of 1 if any owls are left, and 0 if not. Copy this formula down to AB65. In cell AB66, enter =SUM(AB41:AB65)/25. This will give the proportion of runs that the population was still extant after year 25. Finally, in cell J4, make sure the entry is =1-AB66. Create a graph as you did for the Donovan and Welden exercise. When you hit the F9 key, you should get another calculation. Now, compare the following management scenarios. Do this by modifying the appropriate matrix entries. Spotted owls in this case are declining slightly, but you as a manager of a patch of old growth timber want to know what you need to do to best preserve your local population of 25 adult breeding pairs. Some modification of habitat is necessary for fire control. Scenario 1. This management plan involves clearing away some of the understory vegetation that is present in second-growth stands adjacent to the old growth patch where most of the owls are nesting. This will affect the population of wood rats, a major prey species of the spotted owl. Reduce the fertility estimate in cell D4 to 0.1 and cell B5 to 0.05 to reflect reduced reproductive success as a result of reduced woodrat densities. Scenario 2. A timber company has put in a bid to do a large-scale thinning project in the region to reduce the fuel load and subsequent forest fire risk. However, such a plan is not economically feasible unless the timber company is allowed to also cut some of the old-growth patches in the landscape, thus increasing revenues. Although the patches with nesting spotted owls will be protected, it is hypothesized that juvenile owls will have reduced survival during their time as floaters, or non-territorial non-breeders. Reduce C6 to 0.5 to model this reduced survival of the wandering juveniles. Scenario 3. The timber company offers to consider a different old-growth harvest plan, this one leaving some of the outlying patches of old growth to support the non-breeders, but they wish to make up this lost harvest by cutting at the edges of the large patches that support the adult breeders. Reduce adult survival from 0.942 to 0.85 to account for reduced thermal cover and increased likelihood of encountering an aggressive barred owl, which typically results in the spotted owl s death.
For each of these scenarios, recalculate the probability of extinction. Do this 10 times per scenario, entering the results of each run for each scenario in a column for a graph. Which scenario will do the best job of maintaining this population of spotted owls? Graph the probability of extinction for the 10 trials per scenario you ll have three lines, over 10 trials on the x-axis. The other way to do a PVA, the more standard one, is to vary the actual values of the transition matrix (or demographic values for whatever model you use). We ll try this next. Go to the third page in your workbook, type in Spotted Owl Management Options II. On this page, you will create a stochastic matrix, bound it so survival probability remains between 0 and 1 and fertility doesn t drop below zero (no negative owls allowed), then you ll calculate population size through 25 time steps. If the owls disappear by the 25 th time step, they went extinct. You will then compare the same management scenarios using this model. In cells B4, C4,C5, D5, and B6, enter =0. The remaining entries in the matrix will now be means and variances, instead of simply point estimates. In cell D4, type =NORMINV(RAND(),0.226,0.1). The second number is the standard deviation. In cell B5, type =NORMINV(RAND(),0.11,0.03) In cell C6, type =NORMINV(RAND(),0.71,0.2) In cell D6, type =NORMINV(RAND(),0.942,0.15) Hit the F9 key a few times and you may well see survivals that top 1.0 and negative fertilitiesoops! We ll correct this by creating a bounded matrix from this one. In cells F4, G4, G5, H5, and F6, type 0. In cell F5, type =IF($B$5<0,0,$B$5) If the survival probability drops below zero, zero will be entered. In G6, type =IF($C$6>1,1,$C$6) This rate is high enough we won t worry about going negative, but we will worry about exceeding a probability of 1.0, and we ll cap it there. In H6, type =IF($D$6>1,1,$D$6) similar to G5. In cell H4, type =IF($D$4<0,0,$D$4) we ll avoid negative reproduction. In J3, type Start Vector, and in J4-J6, enter 70, 50, and 25 respectively. Set up the following headers: A9: Time Step (create time steps from 0-25 in the column below the header) B9: N0 C9: N1 J9: Trial No. D9:N K9: Results E9: Total Size G9: Extinct?
In B10, enter =$J$4, in C10, enter =$J$5, and in D10, enter =$J$6. E10 =SUM(B10:D10). Finally for this row, in G10 enter =IF(E10>1,1,0) this indicates whether a population is present, or not. In row 11, enter the following: B11: =B10*$F$4 + C10*$G$4 + D10*$H$4 C11: =B10*$F$5 + C10*$G$5 + D10*$H$5 D11: =B10*$F$6 + C10*$G$6 + D10*$H$6 same old matrix multiplication yet again! Copy all of the formulae down to row 35, for the 25 time steps. Then, in G36, enter =IF(SUM(G10:G35)<25,0,1). If population size in any one of the 25 trials drops below zero, the population can t persist (we ll ignore recolonization for this exercise). And, finally, under J9 s header, copy a series of 0-30 down the column to J40. Now, you are going to write a macro (oh, no!) to calculate extinction risk over a series of 30 trials. You can do it manually if you want, but you ll save a lot of time this way and avoid the difficulty of tracking your results. First, go up to Tools-Options-Calculations and switch to manual. Then, go to Tools-Macro-Record New Macro. Assign a letter to your macro and press OK. 1). Press the F9 key. 2). Select cell G36, and copy it. 3). Put your cursor on K9. 4). Go up to Edit, find, search by columns, find next, close. Leave the search field blank. Remember, this is the step that forces the macro to step down to the next available empty cell in your column. 5). The cursor should now be on cell K10. Select Edit, paste special, values, ok. 6). Stop recording your macro. You should now be able to run right down your results column by hitting the macro key (controla for example). If your macro works, write down what the macro command is somewhere in your worksheet for future reference (for example, control-a =macro ). Remember to change the calculation mode back to automatic when you aren t using your macro function. In cell I41, type Extinction Risk. In cell K41, enter =1-(SUM(K10:K40)/30). This is the probability of extinction through the thirty trials of 25 years. You are now ready to compare the scenarios from the previous exercise. Change the mean values in your B4-D6 matrix formulae as required by the scenarios. Re-run your macro to come up with a new probability of extinction for each of the three scenarios. Which of the three scenarios can you recommend? How do these results compare with the previous model, which used a deterministic matrix?
Final Steps: 1. Double the variation in each of your formulae in your B4-D6 matrix. Your formulas should look like: cell D4 =NORMINV(RAND(),0.226,0.2) cell B5 =NORMINV(RAND(),0.11,0.06) cell C6 =NORMINV(RAND(),0.71,0.4) cell D6 =NORMINV(RAND(),0.942,0.3) Go through scenarios 1-3 again with this new level of variation. Re-run your macro to come up with a new extinction risk for each scenario. 2. Now decrease the variation in your parameters to half the original value. Your formulas should look like: cell D4 =NORMINV(RAND(),0.226,0.05) cell B5 =NORMINV(RAND(),0.11,0.015) cell C6 =NORMINV(RAND(),0.71,0.1) cell D6 =NORMINV(RAND(),0.942,0.075) Go through scenarios 1-3 again with your decreased level of variation. Re-run your macro to come up with a new extinction risk for each scenario. Make a table that includes your extinction risk for the three management scenarios for your original model, your model with doubled variation, and your model with halved variation. **What can you conclude about the effects of variation on the outcome of a PVA? **