Matrix models for population management and conservation March 2012 Jean-Dominique LEBRETON David KOONS Olivier GIMENEZ

Lecture 3 Matrix model theory Hal CASWELL, showing a matrix model to a Laysan Albatross. Hal’s book ( Matrix models, Sinauer, 2001) can be used both as a textbook and as a comprehensive reference.

From numerical to formal results v2v2 v1v1 X = 1.05 M V = V M t N 0   (N 0 ) t V asymptotically v 1 V = v 2 … in loose notation

From numerical to formal results t = … M t = … M t./M t-1 = … Termwise division

From numerical to formal results t = … M t = … M t./M t-1 = … Termwise division M t = M M t-1  M t-1, similar to M V = V  M t-1 (and M t ) have columns  proportional to V M t  t u 1 V u 2 V = t VU’ with U’ = u 1 u 2 … in loose notationtranspose

Transposition and matrix product u 1 the transpose of U= is U’= u 1 u 2 u 2 v 1 v 1 u 1 v 1 u 2 if V =, V U’ =, a 2 x 2 matrix v 2 v 2 u 1 v 2 u 2 while U’V = v 1 u 1 + v 2 u 2 is a 1 x 1 matrix, i.e. a scalar, also denoted as  u i v i

From numerical to formal results M t  t u 1 V u 2 V = t V U’ Or, equivalently and more rigorously  -t M t  V U’  u i >0, v i >0 -(t+1) M t+1 = -1 M -t M t  -1 M V U’ = V U’ = -t M t -1 M   V U’  -1 M Hence V U’  -1 M = V U’ Premutiply by U’ and simplify by scalar U’V, to get: U’  M =  U’

Of eigenvalues and eigenvectors M V = V eigenvalue and right eigenvector U’ M = U’ eigenvalue and left eigenvector  -t M t  V U’ leads to M t N 0  t V (U’ N 0 ) asymptotic exponential growth Scalar, weighting the components of N 0 by the u i = « reproductive values » Demographic ergodicity

Of eigenvalues and eigenvectors M V = V eigenvalue and right eigenvector U’ M = U’ eigenvalue and left eigenvector  -t M t  V U’ leads to M t N 0  t V (U’ N 0 ) asymptotic exponential growth Scalar, weighting the components of N 0 by the u i = « reproductive values » Demographic ergodicity, U, V are dispositional properties

Of eigenvalues and eigenvectors M V = V eigenvalue and right eigenvector U’ M = U’ eigenvalue and left eigenvector  -t M t  V U’ leads to M t N 0  t V (U’ N 0 ) asymptotic exponential growth Scalar, weighting the components of N 0 by the u i = « reproductive values » Usually, no formulas, but easy to get numerically

Reproductive values M V = V eigenvalue and right eigenvector U’ M = U’ eigenvalue and left eigenvector  -t M t  V U’ leads to M t N 0  t V (U’ N 0 ) asymptotic exponential growth Scalar, weighting the components of N 0 by the u i = « reproductive values »

Why is it so? These results do not hold for all matrices M is such that M t, for t large enough, has all its terms >0 … because M is a primitive, non negative, irreducible matrix

Why is it so? n x n Matrices have (in general) n eigenvalues which are complex numbers M =

M = Why is it so? n x n Matrices have (in general) n eigenvalues which are complex numbers

Why is it so? However, positive, nonnegative irreducible matrices … M =

Why is it so? However, positive, nonnegative irreducible matrices have their largest modulus eigenvalue which is a positive real number M=

Why is it so? However, positive, nonnegative irreducible matrices have their largest modulus eigenvalue which is a positive real number M= In products such as M t, this “dominant eigenvalue” tends to outweigh the influence of other eigenvalues. i.e., when t   M t N(0)   (0) t V

Of eigenvalues and eigenvectors 1 0 … … 0 0 Eigenvalues are the roots of det (M – … ) = … … 0 1 General Numerical Analysis software (Matlab, Mathematica…) or specialized software (ULM…) will get eigenvalues and eigenvectors for you. Usually, no formulas, but easy to get numerically

Of eigenvalues and eigenvectors The largest root of pf 1 -  pf 2 det ( ) = 2 - (pf 1 +q 2  + pf 1 q 2 - pf 2 q 1 = 0 q 1 q 2 - pf 1 q 2 +  (pf 1 +q 2  2  -4 (pf 1 q 2 - pf 2 q 1 ) is 2

Of eigenvalues and eigenvectors pf 1 q 2 +  (pf 1 +q 2  2  -4 (pf 1 q 2 - pf 2 q 1 ) 2 Even when there is a formula, is not a linear or simple function of the parameters

Of eigenvalues and eigenvectors pf 1 q 2 +  (pf 1 +q 2  2  -4 (pf 1 q 2 - pf 2 q 1 ) 2 Yet, we need to know how varies when one or several parameter values change Even when there is a formula, is not a linear or simple function of the parameters

Sensitivity analysis M =  = What if swallows were not nesting at age 1 ? M =  =

Sensitivity analysis What if ? M  M h = (1-h)M MV= V  (1-h)MV=(1-h) V Hence M h V =(1-h) V   h = (1-h), asymptotic structure V unchanged If you harvest each year 30 % of a roe deer population whose growth rate is 40 % ( =1.4), h is 1.4*(1-0.3) = 0.98, i.e. the population will drop at a rate of 2 % per year What if we harvest a proportion h of a population?

Sensitivity Analysis (  0 ) 00  0 +  (  0 +  ) (  0 )+  In more general cases, can be approximated by a linear function Generic parameter 

Sensitivity Analysis (  0 ) 00  0 +  (  0 +  ) (  0 )+  In more general cases, can be approximated by a linear function Generic parameter  Sensitivity (of wrt  )

Sensitivity and Elasticity Elasticity  Log  Log  = (  )  (  )  (  ) Relative change in vs relative change in  Matrix element  = m ij Lower-level parameter e.g.  = f 1 or  = s 1 Sensitivity  Absolute change in vs Absolute change in 

Sensitivity to matrix element: perturbation analysis  m ij ? M V = V [1] (M+dM)(V+dV) = ( +d ) (V+dV) [2] M V + dM V + MdV + dM dV = V + d V +  dV + d dV [2’] M V + dM V + MdV + dM dV = V + d V +  dV + d dV From [1]: M V + dM V + MdV = V + d V +  dV [2’’] U’ x [2’’]: U’dM V + U’MdV = U’d V +  U’dV U’M=  U’: U’dM V = U’d V i.e. U’dM V = d U’V [3]

Sensitivity to matrix element: perturbation analysis for a change in a single matrix term m ij  m ij +dm ij, 0 0 … 0 … 0 … dm ij … 0 dM = … … 0 0 … 0 hence U’dM V = u i v j dm ij [4]

Sensitivity to matrix element: perturbation analysis  m ij ? U’dM V = d U’V [3] U’dM V = u i v j dm ij [4]  m ij = u i v j / U’V = u i v j /  u i v i a beautiful result due to Hal Caswell (1978)  Log  Log m ij = u i v j / U’V = m ij u i v j /  u i v i

Sensitivity to matrix element: perturbation analysis V = ( v i ), under  v i = 1, is the stable structure m ij u i v j /, under  u i v i = 1, is the relative contribution, in asymptotic regime, of component i to component j, expressed with reproductive value as the currency. As a consequence, elasticities (wrt the m ij ) sum up to 1 Normalization used in general obvious from the context. When speaking of sensitivity, we will use  u i v i = 1 Then,  m ij = u i v j

Sensitivity to lower-level par. the chain rule  ?  =  i  j (  m ij  m ij  ) Barn Swallow example:  s 0 =  m 11  m 11  s 0 +  m 12  m 12  s 0 = u 1 v 1 f 1 + u 1 v 2 f 2 s 0  s 0 = u 1 ( v 1 f 1 s 0 + v 2 f 2 s 0 ) MV = V  v 1 f 1 s 0 + v 2 f 2 s 0 =  v 1, hence the elasticity of wrt s 0 : s 0 / s 0 = u 1 v 1