Predator-Prey
Models
Part 4: Populations as
Functions of Time
We saw in Part 3 that Euler's
Method "works" for systems of differential equations the same
way it does for a single equation -- but it doesn't work very well for
tracking curves that have to loop around and come back where they started.
The "default" differential equation solver in your helper application
uses a more sophisticated technique than Euler's Method, but it's basically
the same idea: Taking small steps in the time domain, predict as accurately
as possible where the next x(t) and y(t) should be. The details
of how to do that we leave to a module in the Differential Equations collection.
But since we know the solver is there, we will use it for the rest of this
module.
The same solver that generates
pairs (x(t),y(t)) in the xy-plane can generate pairs (t,x(t))
in the tx-plane and (t,y(t)) in the ty-plane -- that
is, solutions for the prey and predator populations as functions of time.
Since we know the (x,y) pairs eventually repeat themselves, we must
find x(T) = x(0) and y(T) = y(0) at some time t = T
-- and then the coordinate functions start over through the same values.
That is, x(t) and y(t) are both periodic functions with the
same period T. On the other hand, the trajectories don't look much
like circles or ellipses, so it will not be surprising if the population
functions x(t) and y(t) don't look much like sines and cosines.
- Use the differential equation
solver in your helper application to generate a graph of the prey function
x(t) for about two periods. Then generate a graph of the predator
function y(t) for the same time span, and overlay the two graphs.
What do you notice about the peaks of the two populations?
- Our numerical values for
the coefficients in the Lotka-Volterra model were chosen to roughly approximate
the Hudson Bay data for lynx and hares. Here is the data plot again --
describe in your own words the extent to which Lotka-Volterra is or is
not a good model for this data.
Send comments to the
authors <modules at math.duke.edu>
Last modified: November
11, 1997
