FRWS 6400 10/28/2004
Lab 9 – Harvest and Bootstrapping
Objectives: The first objective of this lab is to try to fit a distribution to l for a population of (guess what) female mule deer. We will be using the data already seen in the midterm, as a good example of real life data and the ensuing issues. That is, it is MESSY. The second objective is to build a population model and run a parametric bootstrap (using the distribution chosen from objective 1) to decide how many more does should be harvested to meet different management goals. In doing this, we will compare fixed harvest with proportional harvest strategies. Finally, time permitting, we will re-run one of our scenarios using a non-parametric bootstrap.
1. Calculate l (i.e., Nt+1/Nt) for the relevant years in the data set and estimate its mean, variance, standard deviation, minimum, and maximum.
a. Build a histogram of the observed lambdas. Fit a scaled beta (example in lab) and normal distribution to the data. Choose one to use in subsequent modeling.
b. Build a discrete, density independent model (for now we will assume that harvest keeps the population in check), that is N(t+1) = N(t)*l. Add temporal stochasticity to your model for l using your chosen distribution. Note, this is a parametric bootstrap. Assume that immigration and emigration are low and not required in the model.
c. Assume harvest occurs after the population processes occur (i.e., population growth® b and d/s). Now harvest an additional 200 females from the population using fixed and proportional harvest strategy.
d. Use a macro to do a simulation of 100 runs (1000 if you have a fast computer or patience!) to record the minimum population size for 1967-1992 for no harvest, fixed harvest, and proportional harvest scenarios. What is the probability that the population size would fall below 4000 (why 4000?) for all 3 scenarios? How do fixed and proportional harvest strategies differ?
e. Copy the worksheet with your population model and simulation runs. Replace the parametric l random variable with a non-parametric version (example in lab). That is, choose l directly from your data – this is a non-parametric bootstrap. Repeat step d. What is the probability that the population size would fall below 4000 for all 3 scenarios? How does this differ from your parametric bootstrap results? Why does it differ?
Extra Credit: In reality, the real reason for modeling this population would probably be to get a feel for how many extra does should have been harvested to reduce the population size about 50% in 5 years (disease control experiment design in mind). What proportion would you recommend harvesting using a proportional harvest strategy? On average, how many does per year is this?
Functions and formulas you will use for this lab:
GEOMEAN(range) – for l
N(t+1) = N(t)l - H – generalized model form
H = Harvest = fixed value or H = N(t)*proportion
NORMINV(probability,mean,std) – temporal stochasticity in l
BETAINV(probability,a,b,min,max) – temporal stochasticity in l
RANDBETWEEN(min,max) – uniform for bounded range, INTEGERS ONLY, “recent” versions (>~1999) of Excel only – else use min+RAND()*(max – min)
VLOOKUP(lookup_value, table_array, col_index_num, range_lookup) – see Excel help too.
VLOOKUP(year, range, offset, false) – lookup a lambda value for given year. Year is randomly chosen via RANBETWEEN, range is the range of your data, offset is how many columns over the l values are from year (counting year as column 1), and false because you don’t want a close value, you only want an exact value.
{=FREQUENCY(B12:B1011,$F$12:$F$18)}
FREQUENCY(data_array,bins_array)
Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A13:A16 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER
=SUM(C11:C1010)/COUNT(C11:C1010) – to calculate probabilities
=ROUND(E15*l,0)
ROUND (number or function, #digits)
2