OVERVIEW

You are required to complete a project on modeling in two parts. In the first part you are to write a mathematical model for a given scenario and study its asymptotic behavior. The second part will have you pick a specific case and carry out linear stability analysis of the model you wrote. The final report will include the description of the model, some calculations, and analysis of the results of the models.

SCENARIO AND PROBLEM STATEMENTS

Consider the dynamics of public opinion about political ideologies.

For simplicity, let’s assume that there are only three options: Republican, Democrat, and Independent. Republican and Democrat are equally attractive (or annoying, maybe) to people, with no fundamental asymmetry between them. The popularities of Republican and Democrat ideologies can be represented by two variables, pr and pd, respectively (0 ? pr ? 1; 0 ? pd ? 1; 0? pr + pd ? 1). This implies that 1- pr – pd = pi represents the popularity of Independent. Assume that at each election poll, people will change their ideological states among the three options according to their relative popularities in the previous poll.

For example, the rate of switching from option X to option Y can be considered proportional to (pY – pX) if pY > pX, or 0 otherwise. You should consider six different cases of such switching behaviors (Republican to Democrat, Republican to Independent, Democrat to Republican, Democrat to Independent, Independent to Republican, and Independent to Democrat) and represent them in dynamical equations.

1. Complete a discrete-time mathematical model that describes this system, and simulate its behavior. See what the possible final outcomes are after a sufficiently long time period. (Hint: Revise Code 4.13 and use.)

For a specific case of pX, pY, and pZ:

1. Find equilibrium points.
2. Calculate the Jacobian matrix at each of the equilibrium points.
3. Calculate the eigenvalues of each of the matrices obtained above. (Hint: Revise Code 5.6 and use.)
4. Based on the results, discuss the stability of each equilibrium point.
5. Why is an all Independent state (electorate) not feasible?
6. Give one major cause of change of state of convergence (Democrat or Republican) from election to election in the context of this model.