1. The problem statement, all variables and given/known data I'm trying to learn the dynamic system model, I was given the following problem to work with. Can someone give me a walk through of did I did right or wrong? Situation: A conservationist is studying a colony of endangered variable oyster-catchers. Birds that are chicks one year will form adult breeding pairs the following year. The probability of a chick surviving to become an adult is 0.3. The probability an adult will survive to breed another year is 0.5. Each year a breeding pair produce on average 2.2 chicks (that is a breeding rate of 1.1 per adult). Initially there are 80 chicks and 20 adult birds. a) Represent this situation by a dynamic system model of the form Xn = A^n X0. You must find matrix A and the initial state vector X0. b) Find X1, the state vector representing the oyster-catcher colony 1 year later. c) Find all eigenvalues of matrix A in part (a) d) By finding the eigenvector corresponding to the numerically largest eigenvalue, estimate the long term ratio of chicks:adults. Hint: If landa is an eigenvalue of matrix A with eigenvector v then A^n v = landa^n v e) Long-term, is this colony of oyster-catchers likely to survive? 2. Known equation A - landa I (to find eigenvalue and eigenvector) |A-landa I| = 0 (to find eigenvector and eigenvalue) AX0=c1Av1+c2Av2 (to find the rate of change after x time) 3. The attempt at a solution a) Chicks n+1 = 0 chicks n + 1.1 adults n Adults n+1 = 0.3 chicks n + 0.5 adults n where X0 = (80; 20) So A= Xn+1 = (0, 1.1; 0.3, 0.5)(chicks n; adult n) therefore Xn = A^n X0 = (0, 1.1; 0.3, 0.5)(80; 20) b) At first I tried to find the eigenvector and eigenvalue by hand, then implied it to the equation A = VDV^-1 to find the value for the long term run. Below is my steps: A-landa I: (0, 1.1; 0.3, 0.5) - landa (1, 0; 0 1) = (0-landa, 1.1; 0.3, 0.1-landa] det (A-landa I) = 0: (-landa)(0.1-landa)-0.33 = 0 expand it: landa^2 - 0.1landa - 0.33 =0 factories: I have got into some trouble factoring with decimal points, so I used matlab to get the eigenvector and eigenvalue which gives: a = 0 1.1000 0.3000 0.5000 >> [v,d]=eig(a) v = -0.9461 -0.7821 0.3238 -0.6232 d = -0.3765 0 0 0.8765 So landa1 = -0.3765, landa 2 = 0.8765 there I put these numbers into the following equation to work out X1 (after one year): X0 = c1*landa 1*v1 + c2*landa 2*v2 = c1 *(-0.9491; 0.3238) + c2 *(-0.7821; -0.6232) = c1 * -0.3765 * (-0.9491; 0.3238) + c2 * 0.8765 *(-0.7821; -0.6232) Since X0 = (80;20) So X1 = 80 * -0.3765 * (-0.9491; 0.3238) + 20 * 0.8765 *(-0.7821; -0.6232) This is the part where I got completely stuck, I knew I did something wrong, I've read the textbook over and over again. But I can't find a solution of where I did wrong, and so I couldn't do the next part. can somebody please help?!