Population Dynamics, single species: In the forest lives a population of rabbits; each rabbit goes through four life stages: newborn, adolescent, mature, and old. Each year, 20% of the newborns live to be adolescent, 50% of the adolescents live to be mature, 80% of the mature rabbits live to be old. Per year, each mature rabbit produces 8 newborns and each old rabbit produces 12 newborns. (At the end of the year, none of the old rabbits survive.) Let Nk, Ak, Mk, and Dk represent the number of newborns, adolescents, mature, and old rabbits, respectively, after k years.(a) Write a system of equations which represents how the population grow each year, (b) Write this system as a matrix equation. (c) What is the largest real eigenvalues? What is the associated eigenvector? (d) As the population stabilizes, approximately how many newborns will there be for each old rab- bit? How many mature rabbits will there be for each old rabbit?