Soay sheep, Ovis aries Island of Soay, viewed from Hirta Images - - PowerPoint PPT Presentation

Soay sheep, Ovis aries

Island of Soay, viewed from Hirta

Images courtesy Soay Sheep Breeders Coop, U of Sheffield

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: * project population dynamics into the future * calculate the long‐run population multiplication rate,  (= the dominant eigenvalue) ( 1) ( ) t t   n An (= the dominant eigenvalue) * calculate the stable stage distribution (= the right eigenvector associated with ) * l l t d ti l ( th l ft i t * calculate reproductive values (= the left eigenvector 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

1 2 k

F F F     

1 2 1 k

P               A L Lefkovitch (1965) first introduced the idea of classifying

1 k

P     

• L. Lefkovitch (1965) first introduced the idea of classifying

individuals by developmental stage, as opposed to age. Thus, more general stage‐structured projection matrices ti ll d L fk it h t i are sometimes called Lefkovitch matrices.

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

T

           A .6 .5   2 .5  

Demographic projection matrices: computing notes F t Ei t b lti li d b ( ) Fact: Eigenvectors can be multiplied by any (non‐zero) constant, and still be an eigenvector. That is, if w is an eigenvector, then ‐2w, w/3, and 1000w 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 d ti l f th ll t t 1 that the reproductive value of the smallest stage = 1. (Remember that reproductive values are relative.)

Demographic projection matrices: sensitivity analysis Th t bl t di t ib ti d d ti l The stable stage distribution w and reproductive value v can 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

i j ij i i i

v w a v w    



i

Demographic projection matrices: sensitivity analysis S iti iti b diffi lt t b Sensitivities can be difficult to compare, because 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  

ij ij ij

a e a      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 eij as the proportional contribution of aij to .