Now suppose that the butterflies that do not migrate reproduce (making one additional butterfly each) and those that do migrate fail to reproduce from exhaustion. No butterflies die.
The above problem extends the migration models to include some reproduction. Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first. Again, xt represents the number of butterflies on the first island, yt represents the number of butterflies on the second island, and pt represents the fraction of butterflies on the first island. In each case, find the following:
a. Start with 100 butterflies on each island and find the number after migration and after reproduction.
b. Find equations for xt+1 and yt+1 in terms of xt and yt.
c. Find the discrete-time dynamical system for pt+1 in terms of pt.
d. Find the equilibrium p*.
e. Sketch a graph and cobweb from a reasonable initial condition.