Consider the dynamics of public opinion about political ideologies. For simplicity, let's assume that there are only three options: Republic-n, 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 S pri 1; 0 S pd 51; [IS pr+ pd S 1). This implies that 1 pr pd 2 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 discretetime mathematical model that descnbes this system, and simulate its behavior. See what the possible nal outcomes are after a sufciently longI time period. (Hint: Revise Code 4.13 and use.) For a specic case of pX, pY, and oz: 2. Find equilibrium points