Example

Draw the vector field for the predator-prey problem

For example, for the point (x,y)= (2,1), we have

If we put more velocity vectors, we get

It is clear that the length of the velocity vectors may affect our understanding of the solutions. If we are only interested by the direction of the motion not its speed, then it is natural to fix a length for these vectors

Compare the two pictures.

Remark. Note that for the predator-prey problem, we may be interested to find out whether one of the two species is facing extinction or not. In other words, we may be interested to study the functions x(t) and y(t) separately. In other words, two more graphs are naturally associated to a system. For the above example, the graph of the solution which satisfies the condition

is drawn below

and the graphs of x(t) and y(t) are

and

Find the equilibrium points of the Duffin system

Answer. We consider the system

Plugging y=0 in the second equation, we get

which gives x = 0, or . Hence the equilibrium points are