Leslie matrices
A Leslie matrix is a discrete-time model for an age-structured population. It combines age-specific fertility and survival into one matrix that projects the population from one time step to the next.
Why use a matrix?
If a population has several age classes, one equation for total population size is not enough. We need to know how many individuals are in each class because fertility and survival can differ strongly with age.
A matrix keeps all these contributions organised and allows the whole age structure to be updated at once.
The population vector
Suppose we divide the population into three age classes. Write
\[\mathbf n_t=\begin{pmatrix}n_1(t)\\n_2(t)\\n_3(t)\end{pmatrix}.\]Here \(n_1(t)\), \(n_2(t)\) and \(n_3(t)\) are the numbers in age classes 1, 2 and 3 at time \(t\).
The total population is
\[N(t)=n_1(t)+n_2(t)+n_3(t),\]but the vector contains more information because it preserves the age composition.
Start from the life cycle
Let \(F_i\) be the expected number of new class-1 individuals produced per individual in age class \(i\) during one time step. Let \(S_1\) and \(S_2\) be the probabilities of surviving from class 1 to 2 and from class 2 to 3.
Write the update equations first
The next first age class receives newborns from all reproductive classes:
\[n_1(t+1)=F_1n_1(t)+F_2n_2(t)+F_3n_3(t).\]The next second age class contains survivors from age class 1:
\[n_2(t+1)=S_1n_1(t).\]The next third age class contains survivors from age class 2:
\[n_3(t+1)=S_2n_2(t).\]The Leslie matrix
These equations can be written compactly as
\[\boxed{\mathbf n_{t+1}=L\mathbf n_t},\]where
\[\boxed{L=\begin{pmatrix}F_1&F_2&F_3\\S_1&0&0\\0&S_2&0\end{pmatrix}}.\]| Part of the matrix | Meaning |
|---|---|
| top row | fertility contributions to the youngest age class |
| sub-diagonal | survival into the next age class |
| other entries | zero in the simplest Leslie model because those transitions are not allowed |
Why matrix multiplication gives the correct update
Consider
\[L=\begin{pmatrix}0&1.2&0.6\\0.5&0&0\\0&0.7&0\end{pmatrix},\qquad\mathbf n_t=\begin{pmatrix}100\\60\\40\end{pmatrix}.\]The first row calculates newborns:
\[0(100)+1.2(60)+0.6(40)=96.\]The second row calculates individuals entering age class 2:
\[0.5(100)=50.\]The third row calculates individuals entering age class 3:
\[0.7(60)=42.\]Therefore
\[\boxed{\mathbf n_{t+1}=\begin{pmatrix}96\\50\\42\end{pmatrix}}.\]Repeated projection
After two time steps,
\[\mathbf n_{t+2}=L\mathbf n_{t+1}=L^2\mathbf n_t.\]After \(k\) time steps,
\[\boxed{\mathbf n_{t+k}=L^k\mathbf n_t}.\]Repeated matrix multiplication therefore projects the age structure forward through time.
Age classes through time
The dominant eigenvalue
Long-term population behaviour is controlled by the dominant eigenvalue of \(L\), usually denoted \(\lambda\).
| Dominant eigenvalue | Asymptotic interpretation |
|---|---|
| \(\lambda\gt1\) | population tends to increase from one time step to the next |
| \(\lambda=1\) | population tends to remain constant in overall scale |
| \(\lambda\lt1\) | population tends to decline |
If the time step is one year, \(\lambda\) can be interpreted as the long-run annual multiplication factor under the model assumptions.
Why an eigenvalue appears naturally
An eigenvector \(\mathbf v\) satisfies
\[L\mathbf v=\lambda\mathbf v.\]If the population has age composition proportional to \(\mathbf v\), applying the Leslie matrix changes only its overall size by the factor \(\lambda\). The proportions among age classes remain unchanged.
Stable age distribution
Under suitable conditions, many different initial age structures eventually approach the same relative age composition. This is the stable age distribution.
It is obtained from the right eigenvector associated with the dominant eigenvalue, after normalising its components so that they sum to 1.
Example of a stable age distribution
Transient behaviour
The dominant eigenvalue describes long-term behaviour, but the early population trajectory can depend strongly on the initial age composition.
A population with few reproductive adults may initially decline even when \(\lambda\gt1\). Conversely, an initially adult-heavy population may grow temporarily even if its eventual long-term trend is decline.
Reproductive value
The dominant left eigenvector gives the reproductive value of the age classes. It measures the relative contribution of an individual in each class to future long-term population growth.
An individual in an age class with high future survival and reproduction may have a larger reproductive value than one in a class with little future contribution.
Sensitivity and elasticity
Because \(\lambda\) depends on the fertility and survival entries of the matrix, we can study how strongly long-term growth responds to changes in each entry.
Sensitivity measures the change in \(\lambda\) produced by a small absolute change in a matrix entry. Elasticity measures the response to a proportional change.
These calculations can identify which age-specific processes are most influential for population growth.
Why this matters in conservation
Suppose adult survival has much greater elasticity than juvenile fertility. Protecting adults might then produce a larger improvement in long-term population growth than increasing the number of births.
This does not mean the same intervention is best for every species. The matrix allows the importance of life-history stages to be calculated from the model.
Leslie matrices and harvesting
Age-selective harvesting can be represented by changing survival or fertility entries. Removing reproductive adults can have a different effect from removing juveniles because different age classes contribute differently to future population growth.
Assumptions of the basic Leslie model
The standard model assumes a fixed time step, known fertility and survival rates, identical demographic behaviour within each age class, and rates that remain constant through time.
Density dependence, environmental variation, migration and demographic randomness are absent unless they are added explicitly.
Leslie versus stage-structured matrices
A Leslie matrix is designed for age classes in which individuals normally progress to the next age class after one time step. Stage-structured models are more flexible because individuals may remain within the same biological stage for several time steps.
That is why stage-projection matrices often contain positive diagonal entries, while the simplest Leslie matrix has zeros there.
Deterministic and stochastic extensions
The standard Leslie matrix projects expected population numbers deterministically. In real populations, survival and reproduction are random. Stochastic versions can allow matrix entries to vary with environmental conditions or treat births and survival as discrete random events.