### Systems of Differential Equations: Models of Species Interaction

Part 2.4: Equilibrium Points

Use the applet to experiment with trajectories for a predator-prey system, notice that there seems to be one set of initial conditions for which the trajectory consists of a single point. At such a point neither population function changes; they are in equilibrium. For this reason such a point is said to be an equilibrium point.

Predator-Prey Direction Field and Trajectories

1. Return to your computer algebra worksheet. For our ongoing example, determine each of the coordinates of the equilibrium point of the system. Your calculations should be accurate to at least two significant digits.

dx/dt = x - 0.03xy

dy/dt = -0.4y + 0.01xy

• (a) Consider the general predator-prey model.

dx/dt = ax - bxy

dy/dt = -cy + pxy

If (x0,y0) is the equilibrium point, explain why x0 and y0 must satisfy

ax0 - bx0y0 = 0

-cy0 +px0y0 = 0

(b) Use the result in (a) to calculate the coordinates of the equilibrium point for the system with the coefficient values a =1, b = 0.03, c = 0.4, and p = 0.01 . Compare the results of this calculation with your calculation in Step 1.

2. Let (x0,y0) be the equilibrium point for the general system

ax0 - bx0y0 = 0

-cy0 +px0y0 = 0
.

The lines x = x0 and y = y0 divide the first quadrant in the xy-plane into four subregions as indicated below.

For each subregion indicate whether x is increasing or decreasing and whether y is increasing or decreasing.

Now we modify our applet once again. Now you have the chance to vary each of the coefficients a, b, c, and p. As you do so, the direction field will change and, of course, the trajectories will change as well.

Predator-Prey Direction Field With Trajectories

1. For each of the four coefficients, a, b, c, and p, experiment with increasing and decreasing that coefficient while leaving the other three alone. When you have finished experimenting with a particular coefficient, return it to roughly the middle of its range. For each coefficient answer the following questions:

(a) How does the equilibrium point change as the coefficient changes?

(b) How does the pattern of trajectories change as the coefficient changes?

(c) Do the changes noted in (a) and (b) agree with your intuition about predator-prey interactions?