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Date Created: 09/17/15
The demography of your invented creature The demography of your invented creature Eigenvalues and eigenvectors Before answering these exercises Study and then run mat2valvecs708m Obtain all the eigenvectors and eigenvalues for your matrix using the MATLAB lnction eig Type help eig to leam how to get all these and to read them Study the example of the turtle demography in the mscript For example if your matrix is named ltltmymatgtgt then you should type ltltVD eigmymat The columns of V will be the eigenvectors and the diagonal of D will be the eigenvalues You will want eigenvalues eigenvectors absolute magnitudes of the eigenvalues and you will want to convert the appropriate vectors to stable stage distribution reproductive values and population growth rates but rst you will investigate them more fully a There is a list of eigenvalues there are more eigenvalues than the dominant one Some are real numbers and some are complex numbers Recall that a complex number is written abi where a is the real part and bi is the imaginary part and i is the square root of 1 Complex numbers occur in conjugate pairs abi and a bi Identify the real and the complex eigenvalues b There are several 39 Each 39 is 39 A with its own eigenvalue Real eigenvalues have real eigenvectors while complex eigenvalues have complex eigenvectors Eigenvectors are directions and eigenvalues are the speeds associated with each direction Both complex and real numbers affect dynamics To understand the dynamics over time one needs to consider how fast the system will move in each direction 2 Write the solution to the projection equation Caswell 2001 Equation 449 for your matrix Don39t try to nd a numerical representation of the 039s just use the symbols for the 0 s The rest of the numbers can be found from a consideration of what is on your printout Give a wordmeaning for the projection equation Save the eigenvalues and eigenvectors of your creature in a mat le so you can call them up again when you need them 3 Investigate eigenvalues that are real numbers in the MATLAB work space you can just work in the command window then write down the results for each lambda suggested here Make a table in which the left hand column is the sequence A 12 l 314 l 5l 6 Across the top of the table enter the following column headings 7t 2 7t 1 7 12 7 12 7 1 7t 2 Fill in the table What happens as 7 is raised to successively higher powers Make some general rules if Lgt1 if 7 1 if is positive but lt1 if is negative but gtl if 1 if ltl do the powers of increase exponentially decrease exponentially exhibit damped oscillation exhibit undamped oscillation exhibit diverging oscillation 4 What about complex numbers Evaluate the behavior of a complex number that is raised to successively higher powers in the MATLAB command window you don t need a le but you should visualize what s happening in the complex plane using the compass command Try the complex number 25i next try the complex number 55i Plot the values of successively higher powers 7 in the complex plane Consider the complex number 7 7 a bi It can also be written in polar coordinates a length and an angle 7 the absolute magnitude of 7 cos6 i sin 19 the absolute magnitude of 7 sq1ta2 b2 and 6 tan 71ba Note that the points can be plotted by their polar coordinates or by their x y a b coordinates The points will be in the same place no matter which of the two coordinate systems is laid out on the space It is just as in the eld you could use a compass and a meter tape or two meter tapes to get to any given point in a plot 5 Study the example of human demography of the United States population of 1966 from Caswell 2001 Example 43 and Figure 46 The angle 9 Caswell s example is given in radians NOTE the number in the table is not 0 it is 971 To get 9 you have to multiply the number in the table by TE Recall that 2 71 radians360 degrees radians 18071 degrees For example 45 degrees 7853 radians 90 degress1 571 radians 180 degrees3141 6 radians 270 degrees 24314 radians 360 degrees 62832 radians 6 Returning to your own creature s matrix analysis call up the eigenvectors and eigenvalues Calculate all the absolute magnitudes and the angles of rotation for all of your eigenvalues The absolute magnitude is given by the square root of a2 b2 The angle is given by tan391ba Plot the eigenvalue spectrum for your matrix in the complex plane gt1 Find the eigenvalue with the largest absolute magnitude and the second largest absolute magnitude Find the real eigenvalue with the largest value dominant eigenvalue This gives the asymptotic population growth rate Compare it to the results you got when you projected your matrix Calculate the damping ratio and discuss the time to convergence Caswell p 95 Discuss brie y the difference between transient dynamics and asymptotic behavior for your matrix 8 Find the 39 39 A with the 39 39 of largest absolute magnitude right eigenvector associated with the dominant eigenvalue Each eigenvector is de ned by the relative proportion of its elements 9 Find the eigenvectors and eigenvalues of the transpose of your matrix Find the dominant 39 39 and its 39 J 39 This one constitutes the left eigenvector of the original matrix associated with its dominant eigenvalue 10 For ease of demographic interpretations demographers normalize vectors to read biological interpretation Normalize the right eigenvector so that its elements sum to l ie so that it can be read as relative proportions of a whole population Make a bar graph of the stable stage distribution Normalize the left eigenvector so that the scalar product of the left and right eigenvectors equals 1 so that its elements are reproductive values Make a line graph of the reproductive values To get these you can use a function ltltsenstuffm It will translate the MATLAB output into these biologically meaningful parameters This is a function le A function m le is different than a script m le A function has input and output arguments There are two ways to call a function that can have several output variables The simple way gives you the rst on the list the other way gives you the full set This function can give you either just one output variable or it can give you 5 output variables These are l the dominant eigenvalue if there is one If there isn t it may give something else 2 the stable stage distribution scaled to sum to one if there is one 3 gives the reproductive values scaled so the scalar product of rv and sdl 4 the sensitivity matrix 5 the elasticity matrix Get all 5 by using the call lam sd rv sen elasenstulTmymatrix Save the output in a ma le so you can use it tomorrow