# Section 4 - With Both Predators and Prey

## What happens when y and x interact?

When x and y interact, the predators (y) feed upon the adult prey (a) and the populations are governed by the equations:

c' = ba - gc - dc

[ bg - d(g + d) > 0 ]
a' = gc - da - uay
[ b, g, d, u, n > 0 ]
y' = - ny + uay

For a given set of parameters, the populations a, c and y will swirl in to a fixed point. To see this, use DStool. (Note you may need to hit continue several times in the orbits window, because it takes quite a while for the populations to stablize.)

Use {b,g,d,u,n} = {2,2,1,1,1}]

First, plot a on the horizontal axis and c on the vertical axis.

[phase portrait]

It is a sink; the limits (final coordinates) of c and a are (respectively) the c- and a-coordinates of the equilibrium which is the center of this sink.

Second, plot a on the horizontal axis and y on the vertical axis.

[phase portrait]

Again, it is a sink; the limit of y is the y-coordinate of the equilibrium

Third, plot c on the horizontal axis and y on the vertical axis.

[phase portrait]

As expected, it is a sink.

It should be noted that the coordinates of this equilibrium are independent of the initial concentrations of c, a, and y. To verify this, check the three phase portraits to see if the center of the sink moves when the path starts at a different spot (point and click).

It doesn't change the center of the sink, although it does change the paths which take it there -- the actual interactions of the two populations are slightly different, but the difference becomes minimal as the system approaches equilibrium.