2. Urban Population Dynamics Population modeling is useful from many different p

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2. Urban Population Dynamics Population modeling is useful from many different perspectives: planners at the city, state, and national level who look at human populations and need forecasts of populations in order to do planning for future needs. These future needs include housing, schools, care for the elderly, jobs, and utilities such as electricity, water and transportation. businesses do population planning so as to predict how the portions of the population that use their product will changing. ecologists use population models to study ecological systems, especially those where endangered species are involved so as to try to find measures that will restore the population. medical researchers treat microorganisms and viruses as populations and seek to understand the dynamics of their populations; especially why some thrive in certain environments but don't in others. The basic equations we begin with are: (1) xk Axk ka 0,1,2,. and x(0) given with solution found iteratively to be (2) xk Akxo We are studying the population dynamics of Los Angeles for the purpose of making a planning proposal to the city which will form the basis for predicting school, transportation, housing, water, and electrical needs for the years from 2000 on. We'll take the unit of time to be 10 years, and use 7 age groups: 0-9, 10-19, 50-59, 60+. Suppose further that the population distribution as of 1990 (the last census is (3.1,2.8, 2.0, 2.5, 2.0, 1.8, 2.9) (x105) This is the xo vector.) and that the Leslie matrix,A, for this model appears as

Explanation / Answer

Using R

codes and outputs

# matrix A as given in the question
> A<- matrix(c(0,.7,0,0,0,0,0,1.2,0,.82,0,0,0,0,1.1,0,0,.97,0,0,0,.9,0,0,0,.9,0,0,.1,0,0,0,0,.9,0,0,0,0,0,0,0,.87,0,0,0,0,0,0,0),nrow=7,ncol=7)
> A
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.0 1.20 1.10 0.9 0.1 0.00 0
[2,] 0.7 0.00 0.00 0.0 0.0 0.00 0
[3,] 0.0 0.82 0.00 0.0 0.0 0.00 0
[4,] 0.0 0.00 0.97 0.0 0.0 0.00 0
[5,] 0.0 0.00 0.00 0.9 0.0 0.00 0
[6,] 0.0 0.00 0.00 0.0 0.9 0.00 0
[7,] 0.0 0.00 0.00 0.0 0.0 0.87 0
> #matrix x0 as given in question
> x0<-matrix(c(3.1*10^5,2.8*10^5,2.0*10^5,2.5*10^5,2.0*10^5,1.8*10^5,2.9*10^5),nrow=1,ncol=7)
> x0
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 310000 280000 2e+05 250000 2e+05 180000 290000
> #population in 2000 is
> x2000<-x0%*%A^10
> x2000
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 7909.307 1946928 988416.2 177826 62762.12 72042.79 0
> #population in 2010 is
> x2010<-x0%*%A^20
> x2010
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 223.4183 11888434 2221474 62004.09 21883.8 17897.12 0
> #population in 2020 is
> x2020<-x0%*%A^30
> x2020
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 6.311015 73587177 5509566 21619.49 7630.408 4446.063 0
> #population in 2030 is
> x2030<-x0%*%A^30
> x2030
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 6.311015 73587177 5509566 21619.49 7630.408 4446.063 0
> #toral population in 2020 is
> sum(x2020)
[1] 79130445
> #toral population in 2030 is
> sum(x2030)
[1] 79130445

Hence the population distribution in 2000 2010 2020 and 2030 is given above as per age. also the total population in year 2020 and 2030 is given and it can be observed that the population total is exactly same which is not realistic and implies that the Leslie matrix is not valid for projection of population after 20 or 30 years as the birth rate and survival rate will change in such a long period

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