Lecture 4

Reproductive Values

The effect of adding a single individual to the initial population at a given age on the asymptotic population size
“present value of future offspring”
Depends on a) the amount of future reproduction, b) the probability of surviving to realize to it, c) the age at which future reproduction will occur.

Note, additional discussion on pedagogy is at (https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.2307/20168257), although I’m not sure I fully follow their logic. Below is confirmed by Otto and Day calculations.

m={{0,1,3,2},{0.5,0,0,0},{0,0.8,0,0},{0,0,0.5,0}};
n={20,30,10,8};

Age	Survival	Survivorship	Fecundity	
0	0.5		1			0
1	0.8		0.5			1
2	0.5		0.4			3
3	0		0.2			2

(* values from example in Otto and Day, page 390 and 392*)(*
m={{0,0,0,0.3},{0.92,0.86,0,0},{0,0.08,0.8,1-0.12},{0,0.02,0.19,0}};
*)

λ=Re[Eigenvalues[m][[1]]]

MatrixForm[m]

Stable age distribution, which is the normalized right eigenvector. It is normalized to sum to zero.

w=((Re[Eigenvectors[m][[1]]])/(Total[Re[Eigenvectors[m][[1]]]]))

Examine the reproductive values of each age class by taking the “left eigenvalues”. In short, you do this in Mathematica by finding the eigenvector of the transpose. Then dividing all by the first age so that the first reproductive value is 1 and others are proportional to the first. You do this because eigenvectors can be scaled arbitrarily, so in this case you scale it by the value of the new-born individual.

v=((Re[Eigenvectors[Transpose[m]][[1]]])/(Re[Eigenvectors[Transpose[m]][[1,1]]]))

v.w

Sensitivity and Elasticity Analysis

Sensitivity: The additive effect of changing a parameter in the projection matrix.

In short, ((dλ)/dz) where z is the parameter of interest 

Elasticity: The multiplicative effect of changing a parameter, or the proportional effect. (z/λ)((dλ)/dz)

The sensitivity matrix will output the sensitivity of all entries in the matrix, even those for which anything but 0 is biologically meaningless.

(Subscript[s, ij])=(((Subscript[v, i])(Subscript[w, j]))/(v.w))

Re[((Transpose[Table[v,{Length[v]}]]Table[w,{Length[w]}])/(v.w))]//MatrixForm

Instead of examining additive changes, the elasticity examines multiplicative changes, ((∂ log(λ))/(∂ log((Subscript[a, ij])))).  This is equivalent to ((Subscript[a, ij])/λ)((∂ λ)/(∂ (Subscript[a, ij]))).

s=Re[((Transpose[Table[v,{Length[v]}]]Table[w,{Length[w]}])/(v.w))];

((s m)/λ)//MatrixForm

Leftover stuff

Here is a set of rules describing the dynamics that result from different values of the dominant eigenvalue :
  
  1 < λ  grows exponentially

0 < λ < 1 decays to zero

-1 < λ < 0 damped period - 2 oscillations as it decays

λ < -1 divergent period - 2 oscillations

complex number :  oscillations

We won' t use all these rules here, but you will use them in the future.

Problem Set

1) Even before you calculate the reproductive values for the Orca model from the earlier problem set, you should be able to guess what the reproductive value will be for the final stage.  Calculate the reproductive values for the Orca model.  Based on this model, if a conservation manager were faced with the choice of saving an adult Orca or a yearling Orca, which should be saved.

2) Calculate the sensitivity matrix of the model.  As a check to be sure the code is working, be sure the sensitivity to the survival rate of post-reproductive females makes sense.  If you were a conservation manager in charge of increasing the population growth rate of the Orca, where would you focus your effort?

3) Calculate the elasticity. Does that change where you would focus your conservation effort?