Question 1:

Consider a population of fish described by the Logistic equation with R0 = 0.1, K=100.

  1. Using R produce a graph of the production function for this population. [2]
# Define parameters
R <- 0.1  # Growth rate
K <- 100  # Carrying capacity

# Generate population sizes
N <- seq(0, K, length.out = 100)

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

  1. Explain in your own words what the production function represents. [2]

    The production function describes the relationship between natural population growth as population increases. The natural population growth reaches a maximum when the population is at half of the carrying capacity, where it then decreases, reaching zero when it reaches the carrying capacity.

    1. For which values of N is the production function zero? [2]

      When N equals 0 or K (100 in this case).

    2. For each value of N, explain from a biological perspective why the production function is zero. [2]

      When N = 0, there is no potential for the population to grow, as there is not a population to begin with. When N = K, the birth and death rates have balanced out due to the effects of crowding, i.e., it is the maximum sustainable size of the population, so growth cannot occur.