The decomposition of the population growth rate matrix can be a little confusing if you’re used to left-to-right multiplication.

From the lecture #2 slides:

$\array{ \text{Spring t} & & \text{Summer} & & \text{Spring t+1} \\ N(t) & M_1 & N'(t) & M_2 & N(t+1) }$

$M_1 = \left [ \array{f_1 & f_2 \\ 1 & 0 \\ 0 & 1} \right ] M_2 = \left [ \array{s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right ]$

## Pre-breeding in t to t+1

We start with the population size vector, $$N_t$$: $N_t = \left[ \array{ N^1_t \\ N^2_t} \right]$

The formula for the exponential population growth rate is:

$N_{t+1} = \lambda \times N_t$

With scaler values, the order of the multiplication is irrelevant $$(\lambda \times N_t = N_t \times \lambda$$).

The formula for the matrix model population growth rate:

$N_{t+1} = M \times N_t$

Notice that the product of N and M goes right-to-left. So, decomposing $$M$$ will also be a right-to-left product.

Decomposing $$M$$ into pre and post breeding:

$N_{t+1} = A_2 \times A_1 \times N_t$

$$A_1$$ is 3 X 2 matrix producing the 3 age-classes just after breeding from the two age=classes just before breeding:

$A_1 = \left[ \array{ f_1 & f_2 \\ 1 & 0 \\ 0 & 1} \right]$

The first post-breeding age-class is produced by the pre-breeding age-classes multiplied by their respective fecundity rates ($$f_1, f_2$$). The second age-class ($$N'^1_t$$) is just the first pre-breeding age-class ($$N^1_t$$) and the third age-class ($$N'^2_t$$) is just the second pre-breeding age-class ($$N^2_t$$).

So, $N'_t = A_1 \times N_t = \left [ \array{ f_1 N^1_t + f_2 N^2_t \\ N^1_t \\ N^2_t} \right ]$

Now, A2 produces the pre-breeding population size vector in time t+1 from the post-breeding vector in time t:

$A_2 = \left[ \array{ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right]$ The first post-breeding age-class consists of the age=0 animals which survived. The second age-class consists of the sum of the post-breeding age=1 and age=2 animals which survived.

And $$N_{t+1}$$ will be:

$N_{t+1} = A_2 \times N'_t = \left[ \array{ s_0 (f_1 N^1_t + f_2N^2_t) \\ s_1 N^1_t + s_2 N^2_t } \right]$

Instead of computing $$N_{t+1}$$ in two steps, we can multiply matrices, $$A_2$$ and $$A_1$$, then get $N_{t+1} = A \times N_t$

where $A = A_2 \times A_1 = \left[ \array{ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right] \left[ \array{ f_1 & f_2 \\ 1 & 0 \\ 0 & 1} \right] = \left[ \array{ s_0 f_1 & s_0 f_2 & \\ s_1 & s_2 \\ } \right ]$

Then, $$N_{t+1}$$ can be computed as just, $$N_{t+1} = A \times N_t$$:

$N_{t+1} = A \times N_t = \left[ \array{ s_0 f_1 & s_0 f_2 & \\ s_1 & s_2 \\ } \right ] \left[ \array{ N^1_t \\ N^2_t} \right] = \left[ \array{s_0 f_1 N^1_t + s_0 f_2 N^2_t \\ s_1 N^1_t + s_2 N^2_t} \right ]$

The result is the same as when we computed it in two steps.

## Post breeding in t to t+1

We start with post-breeding population in time t, $$N'_t$$, a vector with 3 elements, age=0, age=1 and age=2 animals:

$N'_t = \left [ \array{N'^1_t \\ N'^2_t \\ N'^3_t} \right ]$

We can compute the pre-breeding population size in t+1 as:

$N_{t+1} = A_2 \times N'_t = \left[ \array{ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right] \times \left [ \array{N'^1_t \\ N'^2_t \\ N'^3_t} \right ] = \left [ \array{ s_0 N'^1_t \\ s_1 N'^2_t + s_2 N'^3_t} \right ]$

Next, we compute post-breeding population size in t+1, $$N'_{t+1}$$:

$N'_{t+1} = A_1 \times N_{t+1} = \left[ \array{ f_1 & f_2 \\ 1 & 0 \\ 0 & 1} \right] \left [ \array{ s_0 N'^1_{t+1} \\ s_1 N'^2_{t+1} + s_2 N'^3_{t+1}} \right ] = \left[ \array{ f_1 s_0 N'^1_{t+1} + f_2 (s_1N'^2_{t+1} + s_2 N'^3_{t+1}) \\ s_0 N'^1_{t+1} \\ s_1 N'^2_{t+1} + s_2 N'^3_{t+1} } \right]$

We can combine the two transition matrices to compute $$N'_{t+1}$$:

$A = A_1 \times A_2 = \left[ \array{ f_1 & f_2 \\ 1 & 0 \\ 0 & 1} \right] \left[ \array{ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right] = \left[ \array{f_1 s_0 & f_2 s_1 & f_2 s_2 \\ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right]$

Then, $$N'_{t+1}$$ can be computed as $$A \times N'_t$$:

$N'_{t+1} = A \times N'_t = \left[ \array{f_1 s_0 & f_2 s_1 & f_2 s_2 \\ s_0 & 0 & 0 \\ 0 & s_1 & s_2} \right] \left[ \array{N'^1_t \\ N'^2_t \\ N'^3_t} \right] = \left[ \array{f_1 s_0 N'^1_t + f_2 s_1 N'^2_t + f_2 s_2 N'^3_t \\ s_0 N'^1_t \\ s_1 N'^2_t + s_2 N'^3_t} \right]$

The result is the same as when computed in two steps.