Candidates Dewey and Howe are competing for fickle voters. A total of 100,000 people are registered to vote in the election, and each will vote for one of these two candidates. Voters often switch their allegiance. 20% of Dewey's supporters switch to Howe. Howe's supporters are more likely to switch when Dewey is doing well: the fraction switching from Howe to Dewey is proportional to Dewey's percentage of the vote-none switch if Dewey commands 0% of the vote, and 50% switch if Dewey commands 100% of the vote. Suppose Howe starts with 90% of the vote.
a. Find the number of votes Dewey and Howe have after a week.
b. Find Dewey's percentage after a week.
c. Find the discrete-time dynamical system describing Dewey's percentage.
d. Graph the updating function and find the equilibrium or equilibria.
e. Who will win the election?