Question 5 of this section
The population of an endangered hawk species and its natural prey the field mouse, originally 30 hawks and 2000 mice, is represented by the following model: h k+1 = 0.55hk + 0.005mk mk + 1 = -18 hk + 1.2mk where hk and mk represent the hawk and mice population respectively in year k. Make sure you understand what each parameter means. Express the model in the matrix form V k + 1 = AVk, and write V0 = Find the eigenvalues and corresponding eigenvectors of A, use them to diagonalize the matrix A, and write a general formula to calculate Vk. Calculate the predicted population every few years for the next 30 years or so and, using eigenvalues, reason whether an equilibrium will be reached. In 5 years, biologists try to improve the hawk population by introducing 1000 mice in the hawk hunting area. Predict and compare the new hawk population 10 years from now. From that point on, the hawk survival rate drops to 0.50. The biologists are alarmed that this may wipe out the hawk population. Test to see what the model is predicting and explain what is happening. Describe the trajectories of the dynamical system with matrix A = [0 1 -1 0].