1. The Galpagos penguin is the only penguin species found north of the equator, and it...

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1. The Galpagos penguin is the only penguin species found north of the equator, and it is considered endangered. The population of the Galpagos penguin changes on a yearly basis as a discrete dynamic system. Suppose that initially there are 40 chicks and 40 breeding adults, that is zo = 40 Suppose 40 also that the yearly transition matrix is A KE 0 2 0.125 0.75 (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? Which entry tells us the proportion of chicks that will survive to become adults? (b) Calculate the state vector . (c) Find the eigenvalues and corresponding eigenvectors for matrix A. (d) Express z as a linear combination of eigenvectors. (e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults. (f) From your result to part (e), is the Galpagos penguin likely to survive? 1. The Galpagos penguin is the only penguin species found north of the equator, and it is considered endangered. The population of the Galpagos penguin changes on a yearly basis as a discrete dynamic system. Suppose that initially there are 40 chicks and 40 breeding adults, that is zo = 40 Suppose 40 also that the yearly transition matrix is A KE 0 2 0.125 0.75 (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? Which entry tells us the proportion of chicks that will survive to become adults? (b) Calculate the state vector . (c) Find the eigenvalues and corresponding eigenvectors for matrix A. (d) Express z as a linear combination of eigenvectors. (e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults. (f) From your result to part (e), is the Galpagos penguin likely to survive?