The black robbin is the most rare bird in the world

The black robbin is the most rare bird in the world. It is found only in the Chatham Islands, New Zealand. The population of black robbins changes on a yearly basis as a discrete dynamic system. Suppose that initially there are 30 juvenile chicks and 20 breeding adults, that is x0 = 30 20 . Suppose also that the yearly transition matrix is A = 0 1:25 0:75 0:5 : (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? (b) Which entry tells us the proportion of chicks that will survive to become adults? (c) Calculate the state vector x1. (d) Find the eigenvalues and corresponding eigenvectors for matrix A. (e) Express x0 as a linear combination of eigenvectors. (f) Find the long-term distribution, after n years. State the long term ratio of juveniles to adults. (g) From your result to part (f), is the black robbin likely to survive? 2. (20 marks, 4 marks each) (a) State the order of the following DE, and conrm that the functions in the given family are solutions 2dy dx + y = x ?? 1; y = ce??x2 + x ?? 3 (c a constant): (b) Use implicit dierentiation to conrm that the equation x2+xy2 = C denes an implicit solution of the dierential equation 2x + y2 + 2xy dy dx = 0: CONTINUED 1 (c) Solve the separable linear rst order DE dy dx ?? 4xy = 0; using both the method of separation of variables and using an integrating factor.

CLICK HERE TO ORDER NOW