In Example 4, Voting Tendencies of the Political Parties, experiment with starting value...

In Example 4, Voting Tendencies of the Political Parties, experiment with starting values near the origin. Does the origin appear to be a stable equilibrium? Explain. Experiment with different values for the coefficients using the starting values given.

Then try different starting values. What is the long-term behavior?Doyour experimental results indicate that the model is sensitive to the coefficients? To the starting values?

Now assume that each party recruits new party members. Initially assume that the total number of voters increases as each party recruits unregistered citizens. Experiment with different values for new party members. What is the long-term behavior? Does it seem to be sensitive to the recruiting rates? How would you adjust your model to reflect that the total number of citizens in the voting district is constant? Adjust the model to reflect what you think is happening in your voting district. What do you think will happen in your district over the long haul?

REFERENCE:

EXAMPLE 6 An Investment Annuity Revisited

For the annuity modeled in Example 4, how much of an initial investment do we need to deplete the annuity in 20 years (or 240 months)?

Solution The equilibrium value of the system a_n+1 = 1.01a_n −1000 is 100,000 and we want a₂₄₀ = 0. From Theorem 3 we have

and solving this equation gives c = −100,000/(1.01)²⁴⁰ = −9180.58 (to the nearest cent). To find the initial investment a₀, we again use Theorem 3:

Thus, an initial investment of $90,819.42 allows us to withdraw $1000 per month from the account for 20 years (a total withdrawal of $240,000). At the end of 20 years the account is depleted.