Classifying Equilibrium Points -- Answers

1. The equilibrium points are found by solving the equation

   

   Using the notation f(p) = a [1 - ap] p we find the derivative

   

   At zero we see that f'(0) = a so the equilibrium point zero is attracting if |a| < 1.

   For the nonzero equilibrium point we see that

   

   So the nonzero equilibrium point is attracting if 1 < a < 3.

   For population models the constant a is always positive, so for population models we can say the following.

   • If a < 1 there is one biologically significant equilibrium point -- zero. (The nonzero equilibrium point is negative and, hence, of no biological interest.) This equilibrium point is attracting and any population described by such a model will die out.

   • If 1 < a < 3 the zero equilibrium point is not attracting and the nonzero equilibrium point is attracting. For these population models the population "settles down" to the nonzero equilibrium.