Answer

We begin with the blue triangle shown below.

• The hypotenuse

  The direction of migration is parallel to the hypotenuse. That is, the vector field

  F(x, y) = 10 i + 10 j individuals per kilometer per year

  is parallel to the hypotenuse. As a result the rate of migration across this part of the border of the blue triangle is zero.

• The left side

  This side can be described by the vector

  L = 20 j kilometers

  If the migration was perpendicular to this side then the rate of migration across this part of the border of the triangle would be

  ||F(x, y)|| ||L||.

  However, the migration is not perpendicular to the side and only the component of the migration that is perpendicular to the side crosses it. This perpendicular component is

  F(x, y) sin t

  where t is the angle from F(x,y) to L. This leads to the formula

  ||F(x, y)|| ||L|| |sin t|

  for the rate of migration across this side or

  

  Note that the units are individuals per year because we are multiplying migration, measured in units of individuals per year per kilometer by the length of the side measured in units of kilometers and sin t is unitless.

  We can compute this conveniently by using the fact that

  

  The absolute value of this quantity is the rate of migration across the side L. In this case we see that the migration is into the triangle.

• The top

  We can represent the top by the vector

  T = 20 i kilometers

  and compute

  

  So the rate of migration across this side is

  

  individuals per year. In this case we see from the movie that the migration is out of the region. Notice that the emigration across this side exactly balances the immigration across the left side and, since there is no migration across the hypotenuse, the net rate at which the population is changing is zero. Did you expect this? Why?

Now we look at the pink triangle shown in the figure below.

In this triangle the migration across each of the bottom and left sides is

individuals per year and from the movie we see that this is immigration.

The length of the hypotenuse of this triangle is

and, since migration is perpendicular to this side at the rate of 10 individuals per year per kilometer, migration across this side

individuals per year. We see from the movie that this is emigration. Note this emigration exactly balances the immigration across the other two sides.