Could you please help me with this problem on

a) (2 pts) Let M be the Leslie matrix of a 2-dimensional ecological model, and suppose that the eigenvalues of M are A] = 0.8 and 12 =0.5. What is the stability of the equilibrium point at (0, 0)? b) (2 pts) Suppose that A1 = 0.8 has an associated eigenvector U = and 12 = 0.5 has an associated eigenvector V = (2') You know that you can write any initial population vector Xo as a linear combination of U and V. By using these notations, and the ideas you have learned in this course, carefully justify your answer to part a). Your answer should begin with a clear explanation of what the stated stability of (0, 0) means in terms of trajectories of nearby initial conditions. Instead of lengthy verbalizations, use mathematical notation as much as possible.c) (3 pts) Working with the model from parts a) and b), your colleague asks you to analyze the trajectory of the initial population vector XO = 2U +3V. Why do you refuse to do it? d) (3 pts) Now the same colleague asks you to analyze the long-term behavior of a different 2-dimensional ecological model, one whose Leslie matrix M has eigenvalues /1 = -1 and /2 = 2, with associated eigenvectors U = and V = respectively. What do you tell them? Instead of lengthy verbalizations, use mathematical notation as much as possible. (Hint. Start with any population vector Xo = QU + SV. What can you say about a? Now apply M to this linear combina- tion. What can you say about a and B?)