Systems of Differential Equations: Models of Species Interaction
Part 2.3: The Direction Field
Consider again the predator-prey system
dx/dt = ax - bxy
dy/dt = -cy + pxy
Note that the independent variable t does not appear on the right-hand side of either differential equation. This fact will enable us to give another powerful graphical representation of the system.
Consider a trajectory in the xy-plane for a predator-prey system.
Trajectory
If we look at any portion of the trajectory such that the tangent line is never vertical, the trajectory defines y as a function of x.
Portion of the Trajectory
At a particular point on this portion of the trajectory -- the graph of a function y(x), the Chain Rule tells us
dy/dt = (dy/dx) (dx/dt)
Dividing both sides by dx/dt,
we obtain
dy/dx = (dy/dt)/(dx/dt)
But we know both dx/dt and
dy/dt from the differential equations, so
dy/dx = (-cy + pxy)/(ax - bxy)
Now we use the fact that the expression
for dy/dx does not depend on t. Because of this, for any point
in the xy-plane (where dx/dt is non-zero), we can tell what the direction
and slope of any trajectory passing through the point must be.
- Assume a =1, b = 0.03, c = 0.4, and p = 0.01. Find the slope of the trajectory passing through the point (10,20) in the xy-plane.
- At the point in Step 1:
(a) Is x increasing or decreasing?
(b) Is the trajectory being traced out left to right or right to left?
(c) Is y increasing or decreasing?
(d) Is the trajectory being traced out upwards or downwards?
- Suppose dx/dt = 0 but
dy/dt is not 0 at a point in the xy-plane. How can we tell the
direction of the trajectory through this point?
- For any particular predator-prey system, your computer algebra system will display a view of the xy-plane with an array of short arrows indicating the slopes and directions of trajectories through the base points of the arrows. Create such a display for the system with a =1, b = 0.03, c = 0.4, and p = 0.01, and draw several trajectories on top.
The display of short directed line segments you created in Step 3 is called a direction field. You can create a direction field for any system of two differential equations in two dependent variables such that the independent variable does not appear on the right-hand side of either equation. Such systems are said to be autonomous. (You may look at examples of systems that are not autonomous in projects associated with this module.)
The direction field is a powerful tool for understanding the nature of a system of differential equations. Given the direction field, you can sketch in what most of the trajectories look like without any additional computation.
Below we display the same applet you saw in Part 2.2 except that now we have added in the direction field.
Predator-Prey Direction Field and Trajectories
- Experiment with the applet. Before you "click in" the initial conditions, see if you can predict where the trajectory will go.
