Exercise 1:
Estimates for the Newfoundland cod fishery are that K was about 1.4 million tons of fish, and that R0 was about 1 per year.

  1. Given this, what was the MSY and target stock size at MSY (NMSY)?
    \[ \begin{align*} Target\ stock\ size\ &= \frac{K}{2} \\ &= 0.7\ million\ tons \\ &= 700\ thousand\ tons \\ \\ MSY &= \frac{R_0K}{4} \\ &= \frac{1.4}{4} \\ &= 0.35\ million\ tons \\ &= 350\ thousand\ tons \end{align*} \]
  1. Using a pen and a ruler, add two horizontal lines on the graph below, to show i) what was the MSY, ii) what was the target stock size at MSY (NMSY).


  1. Considering the graph and the lines you added, why did the stock decline through the 1970’s? [one sentence should be enough].

Catching was above the MSY for the previous decade, so more fish were being removed from the fishery than natural population growth had the capacity to replenish, causing the stock size to suffer as a result.

  1. Using R produce a graph of the production function for the cod population [HINT: you will need to discount harvest to produce the production function for the Cod population]
# Define parameters
R0 <- 1  # Growth rate
K <- 1400  # Carrying capacity

# Generate population sizes
N <- seq(0, K, length.out = 1400)

# Define logistic growth function
logistic_growth <- function(N, R, K) {
  dN <- R * N * (1 - N / K)
  return(dN)
}

plot(N, logistic_growth(N, R0, K), type = "l", col = "#2970a4",
     xlab = "Population Size (thousands of tons)", 
     ylab = "Natural Population Growth", 
     main = "Production Function for R0=1, K=1400", yaxt="none"
     )

# Adjust y-axis ticks and add grid
grid()
y_ticks <- seq(0,350, by=50)
axis(2, at = y_ticks, labels = y_ticks, las=2)