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Demographic projection matrices A demographic projection matrix is the (square) matrix formed by collecting the average transitions among life ‐ history stages Specifically the element in row i and history stages. Specifically, the element in row i and column j of a demographic projection matrix gives the average transition from stage j to stage i . A demographic projection matrix can be used to:   n ( t 1) An ( ) t * project population dynamics into the future * calculate the long ‐ run population multiplication rate,  (= the dominant eigenvalue) (= the dominant eigenvalue) * calculate the stable stage distribution (= the right eigenvector associated with  ) * * calculate reproductive values (= the left eigenvector l l t d ti l ( th l ft i t associated with  )

Demographic projection matrices: some history Demographic projection matrices for age ‐ structured populations were first popularized by P.H. Leslie (1945), and thus take the name Leslie matrices They take the and thus take the name Leslie matrices . They take the special form    F F F 1 1 2 2 k k      P 0 0     1 A           0 0 P  0  k 1 L Lefkovitch (1965) first introduced the idea of classifying L. Lefkovitch (1965) first introduced the idea of classifying individuals by developmental stage, as opposed to age. Thus, more general stage ‐ structured projection matrices are sometimes called Lefkovitch matrices. ti ll d L fk it h t i

Demographic projection matrices: computing notes U l Unless specified otherwise, the eigenvectors provided by ifi d th i th i t id d b most software are right eigenvectors. Fact: The left eigenvectors of the matrix A are equal to the right eigenvectors of the transpose of A , written A T . Fact: The transpose of a matrix is found by interchanging the rows and columns of the matrix.     0 0 1.5 2 0 .8 0 0         .8 8 0 0 0 0 0 0 0 0 0 0 .7 7 0 0         T A A   0 .7 0 0 1.5 0 0 .6       0 0 0 0 .6 6 .5 5     2 2 0 0 0 0 .5 5  

Demographic projection matrices: computing notes F Fact: Eigenvectors can be multiplied by any (non ‐ zero) t Ei t b lti li d b ( ) constant, and still be an eigenvector. That is, if w is an eigenvector, then ‐ 2 w , w /3, and 1000 w are all eigenvectors as well. Fact: The stable stage distribution w needs to be Fact: The stable stage distribution w needs to be “normalized” so that all of its elements sum to 1. The reproductive value vector v is typically standardized so th t th that the reproductive value of the smallest stage = 1. d ti l f th ll t t 1 (Remember that reproductive values are relative.)

Demographic projection matrices: sensitivity analysis Th The stable stage distribution w and reproductive value v can t bl t di t ib ti d d ti l also be used to calculate sensitivities and elasticities. Loosely, the sensitivity of  to an element of the demographic projection matrix tells us by how much (or how little)  is impacted by changes in that element how little)  is impacted by changes in that element. Mathematically, it is   v w  i j   a v w ij i i i i

Demographic projection matrices: sensitivity analysis S Sensitivities can be difficult to compare, because iti iti b diffi lt t b demographic rates can be measured on very different scales. (See the teasel example.) Solution: Elasticities measure the proportional change in  resulting from a corresponding proportional change in a resulting from a corresponding proportional change in a matrix element. They are defined simply as     a  ij e   ij a ij Fact: The sum of the elasticities across all the elements of a demographic projection matrix equals 1. Therefore, we demographic projection matrix equals 1. Therefore, we can interpret e ij as the proportional contribution of a ij to  .