On the use of matrices in certain population mathematics, P. H. Leslie (1945)

First, a primer on life tables.

https://en.wikipedia.org/wiki/Life_table

Introduction.

This section just explains the way to get estimate values for the female population in any unit of time. For m number of time periods that a population's age structure can be broken up into, you would have to solve m+1 equations to get these different populations for each unit of time. This can be represented by some matrix algebra using the matrix M that they define. The top row of M is the number of daughters that are born to women in a certain age group. The Ps are just the probability of living so that they can be part of the next generation. Thus we get all of the population and how many individuals are in each age group.

Derivation of the matrix elements.

This just explains some of math things that are going on to make the matrix stuff work work. Instead of thinking of the age structures as being continuous from n to n+1, they just assume all people in the age group are n+1/2 to make it easier to run the numbers. Also, since populations should be evenly distributed between n and n+1, the average age of that population would end up being n+1/2 anyways.

Numerical Example.

This gives an example of using the matrix algebra to solve the population numbers for an imaginary population. They use numbers from the life table of a sample species and also do the matrix algebra Mn0 to get the new population structure n1. Here they look at the errors and see that their approximation is actually pretty good with the caveat that the age groups can't be too large in comparison to the life-span of the organism.

Properties of the basic matrix.

This section can be a little confusing, but what they have done is to separate the matrix M into a couple of different parts. It has zeroes in the right upper quadrant because at some point, there will be no more reproduction from those individuals once they get to a certain age. So the submatrix A shows the subpopulation in which reproduction for the organism is occurring. Now they have submatrix A, which is where all of the stuff they are interested in is happening and also it has a determinant and is non-singular and invertible, which makes mathematical operations with it non-trivial.

Transformation of the co-ordinate system.

Once again we get a little weird math stuff going on. They introduce a new vector that is the same as the old n vectors just with different notation (squiggle), xi. Then they introduce another new vector (weird n) eta. This eta is just a made up vector that satisfies the equation eta*xi is equal to the square of the length of the vector. Because it satisfies these things, they can make new vectors, phi (circle with a line) and psi (trident), which can help describe a matrix H such that H*A*H(inverse) is actually the B matrix from the previous page. This B matrix becomes important because it is another way to describe what is happening in the population. Now instead of having the births and likelihood of surviving split up like in matrix A, everything has been collapsed into a single term which is indicated by top row of the matrix.

Relation between the canonical form B and the Lxmx column.

When they are talking about this column, they are talking about life tables again. They are looking at the proportion of surviving individuals from one age group to the next as being the Px value. In reporting these values though, they look at it as if they started with one individual in the 0-1 age group and then look at the proportion surviving on to each age group. Using this process, they can find B with less work and just look at the Lxmx column from the life tables.

The stable age distribution.

Here we get introduced to another matrix algebra concept in the form of eigenvalues (the lambdas). Eigenvalues are important because they can show some characteristics of a matrix as we see, the matrix times a vector is equal to the lambda times the same vector, which makes the job of figuring out solutions of a matrix times a vector much easier. They show that there is a positive eigenvalue and that it will be larger than the others. This eigenvalue becomes the important one as it can show the change in populations represented by xi and psi.

Properties of the stable vectors.

Just some properties of the vector as well as an explanation for why they did it the way they did it.

The spectral set of operators.

Once again they are using the eigenvalues as well as finding a cofactor matrix which they can use to look at the contribution to the total population at time t per individual in a certain age group.

Reduction of B to classical canonical form.

More transformation, making B an easier matrix to use without computers.

The relation between phi and psi vectors.

Not necessarily useful at the time but having to do with solving things in the future. He even says so on page 197.

Case of repeated latent roots.

From here they look at the non-positive eigenvalues that were obtained earlier. They tried to find some of the other roots and whether or not they were important. They found that there are certain cases in which there will be two non-negative eigenvalues though one will be much larger than the other and will thus be dominant. Since it is dominant, the other is pretty trivial.

Special case of the matrix with only a single non-zero Fx element.

Here they are looking at the case where only one age class reproduces. This is an example of what they were doing before but they have actual numbers for this. They show how the matrices A and B end up looking. Because of the numbers, there is a stable population that ends up happening and oscillating between the years but always oscillates in the same pattern. From this example, they take it a little further and change up the numbers since the 6 that are born are born sometime between the 2nd and 3rd year. With this assumption, they find another stable population that tends to occur, but it is a smaller population than before, but it still has the same ratios of individuals in the specific generations and just takes longer to get there.

Numerical comparison with the usual methods of computation

Finally the payoff. Now that they have built this system for solving these equations with less work than before, they can do some comparison to try and work out how well their model fits real data. And that is what they do in this section. This section is probably the most relevant to people and also can be more easily understood than the previous sections. They look at the actual numbers vs the matrix calculated numbers and they go back to the earlier point that the age groups have to be fairly fine in relation to the age of the organism to actually provide meaningful results as they take a look at what happens if you make the age gradient coarser.

Further practical applications

This section is just describing the best situation in which to use these matrices given the time period that this paper was written. It look at new invading species which are important in a lot of ecological systems. These matrices can be used to better anticipate what might happen with these kinds of invasive species or when a niche opens up to a new species. This section also talks about the limitations of these matrices and talks about needing to be able to change with time. This ability to change matrices as they progress through time is something that we can do and is an important facet of modeling biological systems.

Appendix

Check out this section if you want to learn a little bit more about where the got their numbers from in the first 14 sections of the paper. They show the derivations as well as the life tables they were using for their calculations.

Hopefully we will get a chance to talk about some matrix math on Tuesday so that when you read the paper for Thursday you can kind of see what they are doing to some of these matrices.