It has been known for many years that in a predator and prey ecosystem, the numbers of predators and prey fluctuate periodically. For example, red foxes in northern Sweden prey on voles. Studies of these species have shown approximate popThey also eat grouse and hares but appear to prefer voles. Who wouldn’t? ulation cycles in each of the species, with population peaks every 3–4 years. If the number of voles increases there’s more food for the foxes to eat, so their numbers go up, but then they eat a lot of voles so the number of voles goes down, so there’s not enough for the foxes to eat, so the fox numbers go down, so the vole numbers can go back up again, and so on.

Unsurprisingly, real life is far more complicated than this simplistic explanation, but it does provide a starting point for discussion of oscillatory predator-prey cycles.

Probably the most famous example of this oscillatory behaviour is the data set from the Hudson’s Bay Company, tracking the number of snowshoe hare and lynx pelts collected over a 90 year period (Fig. 5.21). Both hare and lynx populations seem to oscillate, with a period of around 10 years.

a. By estimating the maximum and minimum values, the midpoint value and the period, write down a trig function that is an approximate description of the hare data. (It will only be very approximate, of course.)

b. How would you change this trig function to give an approximate description of the lynx data?