# South Africa Population Model

For our application project for this class, we will be analyzing South Africa's population growth, or in this case, the country's population decreases. Based on information from the Central Intelligence Agency's "The World Factbook," South Africa's population growth rate is -0.45% as of 2013. Out of the 233 countries in the world, South Africa's population growth rate is number 222, with number 1 having the largest population growth rate. The country has the twenty-seventh largest population in the world, so we are questioning, why South Africa has a negative population growth rate? What age groups are causing this decline in population?

To figure out the answers to these questions, we are going to use the data from the Central Intelligence Agency’s “The World Factbook,” which has information on every country in the world. This data resource (https://www.cia.gov/library/publications/the-world-factbook/geos/sf.html) provides us with up to date and reliable information on South Africa’s people. We will be using the actual birth rate, death rate, maternal mortality rate, infant mortality rate, and migration rate to analyze the population decline of South Africa. With this data, we will carry out the analysis using the eigenvalue approach to dynamical systems.

To start, we will simplify our data from the Central Intelligence Agency’s “The World Factbook,” to include only five age groups of population. Those age groups will be 0 to 14 years, 15 to 24 years, 25 to 54 years, 55 to 64 years, and 65 years and older. By doing so, for each age category, the population growth rate will be represented by a five-dimensional vector. When we pull all of this information together, we will create a five-by-five matrix that includes the five rates (birth, death, maternal mortality, infant mortality, migration) for each of the age groups of the population. This five-by-five matrix will be our transition matrix (P). Now, we will solve for the eigenvalues by using the equation, determinant(P-I)=0. Once we know the eigenvalues, we will solve for the eigenvectors for each eigenvalue using the equation, (P-nI)x = 0. Each of these eigenvectors will describe the eigenspace for that specific eigenvalue. Since the eigenvectors form a basis for R5, x0=c1v1+c2v2+c3v3+c4v4+c5v5. Then, xk=Pkx0=Pk(c1v1+c2v2+c3v3+c4v4+c5v5). We know that Pkv1=kv1,so xk=c1Pkv1+c2Pkv2+c3Pkv3+c4Pkv4+c5Pkv5=c1kv1+c2kv2+c3kv3+c4kv4+c5kv5.Now, we can input the eigenvalues that we will calculate. To find whether the population growth in South Africa will increase or decrease, we will see what happens when k goes to infinity. If the eigenvalue is larger than one, the population will grow, but if the eigenvalue is less than one, it will decrease. We will find our own population growth rate for South Africa and see what factors are truly causing the population to decline based on the absolute values of our eigenvalues.  In addition, we will ultimately be able to find the ratio of age groups among the population and determine the importance that it holds.