Lab 3 Assignment — Harvest Models
Due by Friday 5:00pm

Answer the following questions and upload your completed Excel file and R script to ELC. Be sure to show your calculations. Undergraduates need to do both exercises in Excel, but only Exercise I in R. Graduate students should do both exercises in Excel and R.

Exercise I

Imagine a population of northern bobwhite (Colinus virginianus) that is experiencing logistic growth with \(r_\mathrm{max} = 0.32\), \(K = 2000\), and an initial population size of 100 individuals.

  1. Project the population for 40 years, and plot abundance over time. Add axis labels as always.
  2. Compute the number of individuals that could be sustainably harvested each year. Plot abundance on the x-axis and sustainable harvest on the y-axis.
  3. At what value of abundance (\(N\)) would maximum sustainable yield (MSY) occur?
  4. What is the value of MSY in this case? In other words, what is maximum number of individuals that can be sustainably harvested?
  5. Using the same values of \(r_\mathrm{max}\), \(K\), and \(N_0\), project the population forward again, but include harvest (\(H_t\)) in the model. Choose values of \(H_t\) that allow for the greatest number of years at MSY. Hint: You can let harvest be zero in some years.

Exercise II

Suppose that annual survival of sitka deer (Odocoileus hemionus sitkensis) decreases as abundance increases according to the equation: \(S = \beta_0 - \beta_1 \times N\).

  1. Compute survival probability for each value of \(N\) provided in the spreadsheet with \(\beta_0=0.95\) and \(\beta_1=0.003\). Create a graph with survival probability on the y-axis and abundance on the x-axis.
  2. A manager is trying to decide how many deer to harvest, and is considering removing anywhere from 10 to 150 individuals from a population of 200. Use the equation above to determine how many deer will remain one year after harvest for each of the harvest options. To accomplish this, do the following:
  • Compute how many individuals will be alive immediately after harvest (Hint: You just have to subtract \(H\) from \(N=200\));
  • Compute survival probability for these remaining individuals using the survival equation above;
  • Compute how many will be alive at the end of the year. In other words, what fraction of the individuals after harvest will survive and be alive next year?
  1. Create a graph with final abundance (\(N\)) on the y-axis and harvest (\(H\)) of the x-axis.
  2. Determine how many deer will be alive if no harvest occurs. Are there any levels of harvest that can result in a larger population than the “no harvest” scenario? If so, how can this be?
  3. If the manager’s objective is to maximize harvest, while maintaining a herd size greater than it would be without harvest, how many deer should be taken?

R tips

Here’s an example of a logistic growth model.

rmax <- 0.1              ## max growth rate
K <- 200                 ## carrying capacity
years <- 2001:2050       ## years
nYears <- length(years)
N1 <- rep(NA, nYears)
N1[1] <- 100             ## abundance in first year
for(t in 2:nYears) {
    N1[t] <- N1[t-1] + N1[t-1]*rmax*(1 - N1[t-1]/K)
}

Here’s an example of a logistic growth model with harvest

h <- 0.02                ## harvest rate (not the sustainable harvest rate)
N2 <- rep(NA, nYears)
N2[1] <- 100             ## abundance in first year
H <- rep(NA, nYears-1)
for(t in 2:nYears) {
    H[t-1] <- N2[t-1]*h  ## harvest
    N2[t] <- N2[t-1] + N2[t-1]*rmax*(1 - N2[t-1]/K) - H[t-1]
}
plot(years, N1, type="l", xlab="Year", ylab="Abundance")
lines(years, N2, col="blue", lty=2)
legend(2000, 200, c("Logistic growth", "Logistic growth with harvest"),
       col=c("black", "blue"), lty=c(1,2))