# How does that change your predictions for 20102020 and 2030? Is the population still unstable?

This project will acquaint you with population modeling and how linear algebra tools may be used to Show more This project will acquaint you with population modeling and how linear algebra tools may be used to study it. Background Kolman pages 305-307. Population modeling is useful from many different perspectives: 1. 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 electricitywater and transportation. 2. businesses do population planning so as to predict how the portions of the population that use their product will be changing. 3. 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. 4.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 dont in others. In human situations it is normal to take intervals of 10 years as the census is taken every 10 years. Thus the age groups would be 0-910-1911-20 etc so 8 or 9 age categories would probably be appropriate. The survival fractions would then show the fraction of newborns (0-9) who survive to age 10 the fraction of 10 to 19 year olds who survive to 20 etc. This type of data is compiled for example by actuaries working for insurance companies for life and medical insurance purposes. The basic equations we begin with are (1) x(k+1) = Ax(k) k=012. . . and x(0) given with solution found iteratively to be (2) x(k) = Akx(0) (see Kolman for details of the structure of A which is 7 x 7 in this case). Your Project Suppose we are studying the population dynamics of Los Angeles for the purpose of making a planning proposal. As above we take the unit of time to be 10 years and take 7 age groups: 0-910-1950-5960+. 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) (x10 5 ) and that the Leslie matrixA for this model appears as Part One: Interpret carefully each of the nonzero terms in the matrix. In addition indicate what factors you think might change those numbers (they might be social economical political or environmental). Part Two: Predict: -what the population distribution will look like in 2000 2010 2020 and 2030 -what the total population will be in each of those years -by what fraction the total population changed each year Additionally what does your software tell you the largest positive eigenvalue of A is? Part Three: Decide if you believe the population is going to zero becoming stable or is unstable in the long run. Be sure and describe in your write up how you arrived at your conclusion. If you have decided it is unstable simulate it long enough that the column matrices for two successive populations are proportional to one another. Calculate that proportionality factor to one decimal place and report it. Part Four: Suppose the birth rates for the second age class can be reduced by 25% by the year 2000. How does that change your predictions for 20102020 and 2030? Is the population still unstable? Show less

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