What ’s important to population growth? A bad question! Good questions are more specific Prospective vs. retrospective questions A parameter which does not vary can not contribute to variation in population growth no matter how high its sensitivity is!

What would happen if Pigs had wings? Pigs’ tails got curlier than they are?

What would happen if …? A prospective question Evolutionary and ecological questions linear perturbation analysis = sensitivity proportional perturbation analysis = elasticity

Analytical entities Dominant eigenvalue Dominant right eigenvector (ssd) Dominant left eigenvector (rv) Derivative of population growth rate with respect to each element in the matrix = sensitivity Derivative of the logarithm of population growth rate with respect to the logarithm of each element in the matrix = elasticity

Analytical procedures Dominant eigenvalue from matrix analysis Dominant right eigenvector (w) from matrix analysis Dominant left eigenvector (v) from matrix analysis

Analytical procedures (cont ’d) For sensitivity, s ij, Multiply elements of left and right dominant eigenvectors w(j) * v(i) For elasticity, e ij, weight these by the ratio of the transition matrix element over the dominant eigenvalue

What has happened in …? Restrospective question  an observed sample of environmental conditions?  a set of fixed treatments?

What influenced variability in population growth in the past? retrospective question How did observed variation in matrices contribute to observed variation in population growth? Main idea: product of sensitivity with variability

A distinct issue: In the context of a temporally varying environment, how does expected co-variation in vital rates influence the difference between the growth rate of a mean matrix and the stochastic growth rate (when both growth rates are expressed on a log scale)?

Retrospective Random treatments: variance decomposition use one mean matrix ’s sensitivity covariance method variance method Fixed treatments: decomposing treatment effects use intermediate matrices ’ sensitivities Regression designs

Do such analyses agree? NO!

What to do? Decide which question you are asking Decide whether you are interested in random variation or in fixed effects Choose the correct reference matrix and analyze its sensitivity Combine a sensitivity measure with a variability measure

Regression design 1 Issue A gradient of conditions, x. How does respond to the gradient at different points; ie, what is the slope at different points? How do different stages contribute to the slope of with respect to x? d /dx =   /  a ij (x)  a ij (x)/  x so: Each contribution is a product of s i j with the slope of a ij wrt to x, these are summed over ij (at each x)

Regression design 2 Steps (see Caswell, 10. 3, p. 273) A regression relationship for each transition matrix element, each a ij depends upon x Use the regressions to generate a series of matrices, A(x) spanning the whole gradient Each A(x) has a sensitivity matrix, S(x) At each x, calculate the contributions of each matrix element calculate d /dx by summing these contributions

Next topics: Disturbance mosaics Megamatrix analysis Stochastic sequence analysis Elasticity of the mean Elasticity of the variability Elasticity by habitat MORE to come