Projection matrix: combining transition and fertility matrices for second stage group

Question

I would like to ask how to combine the following information into the projectio matrix. I do have data for transition (T) and fertility (F) matrices, so the projection matrix (A) is equal to the sum of these two matrices, i.e. A=T+F.

Concerning individuals in transition matrix, they may either move to unit time unit up (become one time unit "older") or down (vice versa) (i.e. entries in transition matrix are either subdiagonal or superdiagonal values).

The question would be how to combine T and F values in A for second stage group, where T and F entries overlap? That is, if probability to move from stage 2 to 3 is equal to e.g. 0.4, and from stage 2 to stage 1 = 0.1, it overlaps with the F value - 3 (which is not a probability, but a different unit). How to meaningully combine this information? Thank you in advance.

Please find below matrix to illustrate the problem. P(i) - probability of moving from one stage to another, and Z(i) - probability of moving from one stage to another backwards (time reversability); F(i) - fertility values.

A= 
    [,1] [,2]           [,3]   [,4] [,5]
[1,] 0.0  F(i)???Z(i)   F(i)   F(i)  F(i)
[2,] P(i)  0.0          Z(i)   0.0   0.0
[3,] 0.0  P(i)          0.0    Z(i)  0.0
[4,] 0.0  0.0           P(i)   0.0   Z(i)
[5,] 0.0  0.0           0.0   P(i)   0.0

Answer 1

Since the expected number of individuals in stage 1 at time t+1 contributed by individuals in stage 2 at time t is indeed F(i)+Z(i) (i.e., individuals can both produce offspring and regress one stage, provided that they reproduce before they regress), the value of A[2,2] should indeed be F(i)+Z(i).