Leslie Matrices

Section 11.2 Leslie Matrices

Subsection 11.2.1 Age-Structured Populations

One of the starting points of mathematical biology is the study of population growth. Though many of the most interesting systems involve multiple species with various interactions (symbiotic, competition, predator-prey, etc.), the theory starts with single-species dynamics. How does a species grow or decline when considered by itself? For those who’ve taken calculus with me, I used population growth as a major motivator for that course, working with the differential equations that describe exponential and logistic growth.

Sometimes I want to know more details about the population than simply the count of its members. One way of describing a population is by age categories; this is called an age-structured population. This is pretty common in human geography, where the population of a country (or other human groups) is given in ten-year increments. (Usually, these graphs are segregated by sex as well.) This can also be very useful in animal populations since there are often important age-defined subsets that control the growth or decline of a population. In conservation biology, resources can be much more carefully used if a model can focus on a specific age category in a population.

In the simplest single-species models, there is a birth and death rate. For an age-structured population, the situation is more complicated. Instead of just asking for the death rate, I now need to know the survival rate from each age category to the next (assuming that no one survives from the oldest age category). If there are \(n\) age categories, that’s \((n-1)\) survival rates (again, excluding the oldest age category). Likewise, instead of a generic birth rate, I need to know the birth rate due to each age category. These are called fecundity rates. There may be fewer of these (or, rather, some might be set to zero) since only certain age categories may be fertile, But again, in theory, I have up to \(n\) new rates.

How do I track all this information? I can do it with the dynamical system described in the next section.

Subsection 11.2.2 Leslie Matrices

A Leslie matrices is a dynamical system that models an age-structured population. To demonstrate, I’ll assume there are four age categories. The state vector \(v_k\) is the current population in each age category: so that \(v_1\) is the population of the youngest category, \(v_2\) of the next, \(v_3\) of the next, and finally \(v_4\) of the oldest category. The matrix \(C\text{,}\) called a Leslie matrix, is a \(4 \times 4\) matrix, and the action of a matrix (like for all dynamical systems) takes a state vector and moves it ahead by one time step. If I use coordinates \(w,x,y,z\) in \(\RR^4\text{,}\) here is the dynamical system.

\begin{equation*} \begin{pmatrix} w_{k+1} \\ x_{k+1} \\ y_{k+1} \\ z_{k+1} \end{pmatrix} = \begin{pmatrix} f_1 \amp f_2 \amp f_3 \amp f_4 \\ s_1 \amp 0 \amp 0 \amp 0 \\ 0 \amp s_2 \amp 0 \amp 0 \\ 0 \amp 0 \amp s_3 \amp 0 \end{pmatrix} \begin{pmatrix} w_k \\ x_k \\ y_k \\ z_k \end{pmatrix} \end{equation*}

Nine of the entries must be set to zero, since they are age categories that cannot transfer to other age categories in one time step (i.e., something in age category 1 cannot jump directly to age category 4, nor can something in the 3rd age category suddenly get younger and move into the 2nd age category). I want to interpret the remaining 7 non-zero entries. The first row outputs to the first category, representing the creation of new members of the population. Each \(f_i\) is the fecundity of each population state: its rate of producing offspring. Of course, some of these might be zero if only certain age categories produce offspring. The \(s_i\text{,}\) on the other hand, are transitions from one age category to the next. These are survival rates: \(s_1\) is the rate of survival from category \(1\) to category \(2\text{,}\) \(s_2\) from category \(2\) to \(3\) and \(s_3\) from category \(3\) to \(4\text{.}\)

These Leslie matrices are irreducible (recall that irreducible meant that each state connects to the other; by survival and fertility, it is possible to pass from any age category to another over several years). This means that there is a unique largest eigenvalue \(\lambda\) with a positive eigenvector. This is the value I care about: in the long run, the largest eigenvalue dominates, and its eigenvector gives the stable age distribution. If \(\lambda = 1\text{,}\) I expect a stable population. If \(\lambda > 1\text{,}\) I expect exponential growth, and if \(\lambda \lt 1\text{,}\) I expect exponential decay. In addition, I can look for the limit of the ratios between the age categories. As the population either grows or decays, those ratios will approach fixed values, which are represented by the eigenvector matching the dominant eigenvalue.

Subsection 11.2.3 Examples of Leslie Matrices

All these examples will be four-stage population models, using the description as above. For each, I will determine the long term behaviour by analyzing the dominant eigenvalue and its matching eigenvector.

Example 11.2.1.

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 0.7 \amp 0.6 \\ 0.8 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.6 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.7 \amp 0 \\ \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 0.83 \amp \amp v = \begin{pmatrix} 1.72 \\ 1.65 \\ 1.19 \\ 1 \end{pmatrix} \end{align*}

The popluation is not viable, since the dominant eigenvalue is less than 1; the population will decay. The age ratios will approach \(1.72:1.65:1.19:1\) from youngest to oldest.

Example 11.2.2.

This example has only changed one fecundity coefficient and one survival coefficient from the previous example. What is the effect?

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 1.1 \amp 0.6 \\ 0.8 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.9 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.7 \amp 0 \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 1.03 \amp \amp v = \begin{pmatrix} 2.15 \\ 1.68 \\ 1.49 \\ 1 \end{pmatrix} \end{align*}

The population is viable; since the dominant eigenvalue is larger than one, the population will grow. The age ratios will approach \(2.15: 1.68:1.49:1\text{.}\) Compared with the previous example, two coefficients have been changed: the fecundity coefficient of the 3rd age category and the survival coefficient from the 2nd to the 3rd age category. These were both increased sufficiently to change the non-viable population into a viable population.

Example 11.2.3.

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 2 \amp 1 \\ 0.4 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.9 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.9 \amp 0 \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 1.01 \amp \amp v = \begin{pmatrix} 3.21 \\ 1.27 \\ 1.16 \\ 1 \end{pmatrix} \end{align*}

This is a viable population; since the dominant eigenvalue is positive, this population will grow. That said, since the dominant eigenvalue is only slightly above 1, the population growth will be relatively slow. As the population grows, the age ratios will approach \(3.21:1.27:1.16:1\text{.}\)

Example 11.2.4.

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 150 \amp 50 \\ 0.03 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.9 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.9 \amp 0 \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 1.63 \amp \amp v = \begin{pmatrix} 196.37 \\ 3.50 \\ 1.87 \\ 1 \end{pmatrix} \end{align*}

The population is viable, since the dominant eigenvalue is larger than 1. In addition, this is quite a fast growth rate, since the eigenvalue is significantly above one. The age ratios will approach \(196.37:3.50:1.87:1\text{.}\) Note that these ratios are quite different from previous examples: this is a population with many many young members, very few of which reach maturity.

Example 11.2.5.

This question has reduced the fecundity compared with the previous example.

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 50 \amp 30 \\ 0.03 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.9 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.9 \amp 0 \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 1.25 \amp \amp v = \begin{pmatrix} 79.63 \\ 1.92 \\ 1.38 \\ 1 \end{pmatrix} \end{align*}

The population is viable, since the dominant eigenvalue is greater than 1. The age ratios will approach \(79.63:1.92:1.38:1\text{.}\) Even with the reduced fecundity coefficient (significantly reduced!), the population is still viable.

Example 11.2.6.

This example has reduced the first survival coefficient compared to the previous example. What is the effect?

\begin{equation*} \begin{pmatrix} 0 \amp 0 \amp 50 \amp 30 \\ 0.01 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0.9 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0.9 \amp 0 \end{pmatrix} \end{equation*}

Solution.

Here is the dominant eigenvalue and the matching eigenvector (calculated by computer).

\begin{align*} \amp \lambda = 0.90 \amp \amp v = \begin{pmatrix} 89.012 \\ 0.993 \\ 0.996 \\ 1 \end{pmatrix} \end{align*}

The population is not viable, since the dominant eigenvalue is less than 1. The age ratios will approach \(89.012:0.993:0.993:1\text{.}\) Compared to the previous example, the viable population has now become non-variable. The only coefficient that has changed is the survival coefficient from the 1st to the 2nd age category. Previously, 3% of the youngest survived for a time interval. Now, only 1% survive. That change is significant enough to drop the population from viable to non-viable.

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