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Dynamic system model 
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#1
Apr2410, 01:13 AM

P: 95

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 oystercatchers. 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 oystercatcher 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) Longterm, is this colony of oystercatchers likely to survive? 2. Known equation A  landa I (to find eigenvalue and eigenvector) Alanda 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: Alanda I: (0, 1.1; 0.3, 0.5)  landa (1, 0; 0 1) = (0landa, 1.1; 0.3, 0.1landa] det (Alanda I) = 0: (landa)(0.1landa)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?! 


#2
Apr2410, 04:57 AM

PF Gold
P: 960

In b) you only need to calculate the population after one year, i.e. with n = 1 calculate x_{1} = A^{1} x_{0} = A x_{0}. No need for eigenvalues yet.
Also, your characteristic equation (before you start using matlab) is not entirely correct. 


#3
Apr2410, 05:36 AM

P: 95




#4
Apr2410, 06:13 AM

PF Gold
P: 960

Dynamic system model
And, just for your information, the Greek letter is called lambda, not landa. 


#5
Apr2410, 06:24 AM

P: 95

yes whenever I'm doing excerise or pratice I always do the question first manually before I check it with matlab, the only time I don't do manually it anymore is when I got so used to working with those type of questions that I could mentally think the answer out, and then I check it with matlab. And yes it was I typo, ahaha! It is 0.5 instead of 0.1



#6
Apr2410, 06:37 PM

P: 95

So x_{1}= C_{1}(0.3765)^1(0.8481; 0.3238) + C_{2}(0.8765)^1(0.7821;0.6232) but since lambda 2 is the domain value it will overrun lambda 1 in the long term, therefore the state vector after 1 year will be 0.8765x_{k}? or did I think too much, the state vector is simply : x_{1}= C_{1}(0.3765)^1(0.8481; 0.3238) + C_{2}(0.8765)^1(0.7821;0.6232) 


#7
Apr2510, 12:51 AM

PF Gold
P: 960

If this is for b) then you are still getting ahead of yourself. You only need to perform a simple matrix multiplication of A and x_{0} to get x_{1}. In this model this is, by definition, how the state for one generation is advanced to the state for the next generation. If you were to calculate, say, the 10th generation you would have to do the multiplication 10 times to get to x_{10}. By the way, it can be instructive to plot how the state develops over time, especially in order to prepare yourself for part d) and e).



#8
Apr2510, 08:49 PM

P: 95

so say if I want to work out _{10}, would I do this: [0^10,1.1; 0.3,0.5^10]*[80,20] 


#9
Apr2610, 01:53 AM

PF Gold
P: 960

Your questions seem to indicate you have missed something fundamentally somewhere in your textbook. If the above explanation is all news to you, I recommend that you take some time to read up on the parts in your textbook that are relevant for this problem or you will most likely have a very tough time understanding the eigenvalues and eigenvectors. 


#10
Apr2610, 05:48 AM

P: 95

so for x_{10} would I do the following? A = VDV^1 [0 1.1; 0.3 0.5]^10 = [0.9461,0.7821; 0.3238,0.6232]*[0.3765^10, 0; 0,0.8765^10]*[0.7394, 0.9279; 0.3842, 1.1225] which gives: [0.0805, 0.2349; 0.0641, 0.1872] since X_{10} = A^10 x_{0}, then: [0.0805, 0.2349; 0.0641, 0.1872]*[80;20] which gives: 11.1340 8.8692 is this correct? 


#12
Apr2710, 03:15 AM

P: 95

yay! thanks for your help



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