Systems of Differential Equations: Models of Species Interaction
Part 1.1: Introduction
In this module our main examples of systems of differential equations will be models of interaction between two species. The model we will consider first assumes that there are exactly two species, one of which -- the predators -- eats the other -- the prey. Such pairs exist throughout nature:
- lions and gazelles,
- birds and insects,
- pandas and eucalyptus trees,
- Venus fly traps and flies.
To keep our model simple,
we will make some assumptions that would be unrealistic in most of these
predator-prey situations. Specifically, we will assume that
- the predator species is
totally dependent on a single prey species as its only food supply,
- the prey species has an
unlimited food supply, and
- there is no threat to the
prey other than the specific predator.
Very few such "pure"
predator-prey interactions have been observed in nature, but there is a
classical set of data on a pair of interacting populations that come close:
the Canadian lynx and snowshoe hare pelt-trading records
of the Hudson Bay Company over almost a century. The following figure (adapted
from Odum, Fundamentals of Ecology, Saunders, 1953) shows
a plot of that data.
There was apparently nothing keeping the hare population in check other
than predation by lynx, and the lynx depended entirely on hares for food.
To be sure, trapping for pelts removed large numbers of both species from
the populations -- otherwise we would have no data -- but these numbers
were quite small in comparison to the total populations, so trapping was
not a significant factor in the size of either population.
On the other hand, it is reasonable to assume that the success of
trapping each species was roughly proportional to the numbers of that species
in the wild at any given time. Thus, the Hudson Bay data give us a reasonable
picture of predator-prey interaction over an extended period of time.
- One striking feature of the two
populations is that they fluctuate periodically. List any other features that
you notice.
-
(a) On average, what was the
period of oscillation of the lynx population?
(b) On average, what was the
period of oscillation of the hare population?
(c) On average, do the peaks
of the predator population match or slightly precede or slightly lag those
of the prey population? If they don't match, by how much do they differ?
(Measure the difference, if any, as a fraction of the average period.)
To be candid, things are
never as simple in nature as we would like to assume in our models. In
areas of Canada where lynx died out completely, there is evidence that
the snowshoe hare population continued to oscillate -- which suggests that
lynx were not the only effective predator for hares. In fact, there is evidence that owls also prey extensively on the hares. However, we will ignore
that in our subsequent development.
