Problem:

Steps:

Solve this problem with following 5 steps please.

Consider a population in a controlled environment (no predators, no epidemics, no shortage of food supplies, unlimited space, no migration). We suppose that the number of individuals changes in time only due to births and deaths. The population considered lives at most three years (3 age groups). Write the mathematical model describing the population dynamics, when we assume that only individuals who are one year old and two years old can procreate. Choose reasonable values for the birth rates and death rates. Choose the initial distribution of individuals in the population. Compute by direct calculation the composition of the population as well as the overall population after one year and after two years. Based on these preliminary results, how do you expect the population to evolve in time? Compute the distribution of population after 100 years. You may need to use eigenvalues and eigenvectors. To speed up the computations, you can use any program you like to compute them (no need to do any computation by hand). Just make sure that it is clear how you get an eigenvalue problem and how to use it to find the population distribution after 100 years. Do not skip steps. Plot the evolution in time of each age group as well as the overall population. Use any program you like and if you do not know how to use any, plot the graphs by hand. Make sure that the initial population actually corresponds to the one you chose. Now suppose that every year there is an exchange with a nearby community: 20% of the individuals who are two years old leave to join the nearby community. How does the model change? Compute the distribution of population after one year and after two years by direct computation. Consider a population in a controlled environment (no predators, no epidemics, no shortage of food supplies, unlimited space, no migration). We suppose that the number of individuals changes in time only due to births and deaths. The population considered lives at most three years (3 age groups). Write the mathematical model describing the population dynamics, when we assume that only individuals who are one year old and two years old can procreate. Choose reasonable values for the birth rates and death rates. Choose the initial distribution of individuals in the population. Compute by direct calculation the composition of the population as well as the overall population after one year and after two years. Based on these preliminary results, how do you expect the population to evolve in time? Compute the distribution of population after 100 years. You may need to use eigenvalues and eigenvectors. To speed up the computations, you can use any program you like to compute them (no need to do any computation by hand). Just make sure that it is clear how you get an eigenvalue problem and how to use it to find the population distribution after 100 years. Do not skip steps. Plot the evolution in time of each age group as well as the overall population. Use any program you like and if you do not know how to use any, plot the graphs by hand. Make sure that the initial population actually corresponds to the one you chose. Now suppose that every year there is an exchange with a nearby community: 20% of the individuals who are two years old leave to join the nearby community. How does the model change? Compute the distribution of population after one year and after two years by direct computation