Consider a model of migration that includes three regions: City, Suburbs, and Rural. Suppose that each decade 15% of city dwellers move to the suburbs, and an additional 10% move to rural areas; 20% of suburban residents move to the city and another 15% move to rural areas; 10% of those living in rural areas move to the city and another 10% to the suburbs; the remainder of the total population remain where they are. (a) Draw a transition diagram to express this problem. (See dolphin problem in lecture slides for an example).
(b) Write out the difference equations.
(c) Give the transition (projection) matrix
(d) Initially, 30,000 people live in the city, none live in the suburbs, and 70,000 people live in rural areas. How many people live in each area after four decades?
(e) Use R to calculate the eigenvalues and eigenvectors of the transition matrix and use the principal eigenvector to describe the relative proportions of people living in the city, suburbs and rural areas in the long-term .