A team of biologists studied the relationship between the rabbit and wolf populations in a certain region. They note that the populations of the animals appear to vary periodically over time. Their observations are as follow:

a) For the wolves, a maximum population of 7,000 occurs 8 months into the study. A
minimum population of 3,000 occurs 12 months later.
b) For the rabbits, a minimum population 10,000 occurs 14 months into the study. The rabbit population was at its maximum of 40,000 two months after the study began.

  1. Write equations that model the rabbit population, r(t), and the wolf population, w(t), where t is months since the start of the study.
  2. Explain why you chose a sine or cosine model for each function.
  3. What are the average populations of rabbits and wolves? Justify your results.
  4. Graph each function, on the same set of axes, over the course of two years. Label all critical points.
  5. Explain the population trends, for each species, for each quarter of the cycle.
    Identify the apparent cause of these trends.

Sample Solution