Age-structured populations
In many populations, age changes survival, reproduction and future contribution to population growth. A model that keeps only the total population size can therefore miss important biological information.
Why total population size is not enough
Suppose two populations both contain 1000 individuals. In the first population, most individuals are juveniles. In the second, most are reproductive adults.
The totals are identical, but their future growth can be very different because the age compositions differ.
Represent the population by an age vector
Suppose we use three classes:
| Class | Meaning |
|---|---|
| 1 | juveniles |
| 2 | young reproductive adults |
| 3 | older adults |
At time \(t\), write
\[\boxed{\mathbf n_t=\begin{pmatrix}n_1(t)\\n_2(t)\\n_3(t)\end{pmatrix}}.\]Each component records the number of individuals in one age class.
The two main processes
Most simple age-structured population models are built from two processes:
Individuals can survive from one age class into the next, and reproductive classes can contribute newborns to the first class.
A simple life-cycle diagram
Survival probabilities
Let
\[s_1=\text{probability a juvenile survives into class 2},\] \[s_2=\text{probability a class-2 adult survives into class 3}.\]If \(n_1(t)\) juveniles are present at time \(t\), then the expected number entering class 2 one time step later is
\[s_1n_1(t).\]Likewise, the expected number entering class 3 is
\[s_2n_2(t).\]Fertility rates
Let
\[f_1,f_2,f_3\]denote the expected numbers of new class-1 individuals contributed per individual in each class over one time step.
Often juveniles do not reproduce, so \(f_1=0\), while adult classes may have positive fertility.
Write the update equations
For the three-class example,
\[n_1(t+1)=f_1n_1(t)+f_2n_2(t)+f_3n_3(t),\] \[n_2(t+1)=s_1n_1(t),\] \[n_3(t+1)=s_2n_2(t).\]These three equations can be written much more compactly using a matrix.
The Leslie matrix
The standard discrete age-structured model is
\[\boxed{\mathbf n_{t+1}=L\mathbf n_t},\]where
\[\boxed{L=\begin{pmatrix}f_1&f_2&f_3\\s_1&0&0\\0&s_2&0\end{pmatrix}}.\]This is called a Leslie matrix.
What each part of the matrix means
| Matrix location | Interpretation |
|---|---|
| top row | fertility: contributions to newborns |
| sub-diagonal entries | survival from one age class to the next |
| zeros elsewhere | transitions not included in this simple age-class model |
How the matrix multiplication works
Suppose
\[L=\begin{pmatrix}0&1.2&0.6\\0.5&0&0\\0&0.7&0\end{pmatrix},\qquad\mathbf n_t=\begin{pmatrix}100\\60\\40\end{pmatrix}.\]Then the juvenile class at the next step is
\[n_1(t+1)=0(100)+1.2(60)+0.6(40)=96.\]The young-adult class is
\[n_2(t+1)=0.5(100)=50,\]and the older-adult class is
\[n_3(t+1)=0.7(60)=42.\]Therefore
\[\boxed{\mathbf n_{t+1}=\begin{pmatrix}96\\50\\42\end{pmatrix}}.\]Why this is biologically useful
The total population has changed from
\[100+60+40=200\]to
\[96+50+42=188.\]But the age structure also changed. That composition matters for what happens next.
Repeated multiplication projects the population forward
After two time steps,
\[\mathbf n_{t+2}=L\mathbf n_{t+1}=L^2\mathbf n_t.\]After \(k\) steps,
\[\boxed{\mathbf n_{t+k}=L^k\mathbf n_t}.\]This is why matrix methods are central to structured population modelling.
Age classes through time
The dominant eigenvalue
For a time-invariant Leslie matrix, long-term population growth is governed by its dominant eigenvalue, usually denoted \(\lambda\).
| Dominant eigenvalue | Long-term interpretation |
|---|---|
| \(\lambda>1\) | population tends to grow from one time step to the next |
| \(\lambda=1\) | population is stationary in overall scale |
| \(\lambda<1\) | population tends to decline |
If the time step is one year, \(\lambda\) can be interpreted as the asymptotic annual multiplication factor under the model assumptions.
Why an eigenvalue appears
An eigenvector \(\mathbf v\) satisfies
\[L\mathbf v=\lambda\mathbf v.\]This means that if the population has age structure \(\mathbf v\), one time step multiplies every component by the same factor \(\lambda\). The proportions among age classes therefore remain unchanged.
Stable age distribution
Under suitable conditions, repeated application of the same Leslie matrix causes many different initial age structures to approach the same relative age composition.
This limiting composition is determined by the positive eigenvector associated with the dominant eigenvalue.
Transient dynamics
Before the stable age distribution is reached, the population can behave quite differently from its long-run trend. This period is called the transient phase.
A population may temporarily decline even when \(\lambda>1\), or temporarily increase even when \(\lambda<1\), depending on its initial age composition.
Reproductive value
The dominant left eigenvector of the Leslie matrix gives reproductive values. These measure how much an individual in each age class contributes, in expectation, to future long-term population growth.
An individual in a highly reproductive class may therefore have greater reproductive value than an individual in a class with low survival or little future reproduction.
Sensitivity and elasticity
Because the long-term growth factor \(\lambda\) depends on the entries of the Leslie matrix, we can ask how strongly \(\lambda\) changes when a fertility or survival parameter changes.
Sensitivity measures the change in \(\lambda\) per absolute change in a matrix entry. Elasticity measures proportional change and is often useful when comparing parameters with different units or scales.
These ideas can identify which life-history stages have the greatest influence on long-term population growth.
Why this matters in conservation
Suppose improving juvenile survival has little effect on \(\lambda\), while improving adult survival has a large effect. A conservation programme may then achieve more population benefit by protecting adults.
Structured models therefore help move beyond the question “how many individuals are there?” to “which demographic processes drive persistence or decline?”
Age structure and harvesting
Harvesting individuals from different age classes can have very different effects. Removing juveniles, reproductive adults or older adults changes different entries or components of the age-structured system.
This is one reason a harvest policy based only on total abundance may miss important demographic consequences.
Age structure in epidemiology
Age structure also matters in infectious-disease modelling because contact patterns, susceptibility, vaccination, infectiousness and severe-disease risk can vary with age.
In epidemic models, age groups are usually coupled through a contact matrix rather than only through ageing transitions, but the general principle is the same: the population is heterogeneous, so one total state variable is insufficient.
Discrete age classes versus continuous age
The Leslie model uses discrete age classes and discrete time steps. Another approach treats age as a continuous variable and describes population density as \(n(a,t)\), where \(a\) is age and \(t\) is time.
This leads to age-structured partial differential equations such as the McKendrick–von Foerster framework. The discrete matrix model is often easier to introduce and compute, while continuous-age models can represent age more finely.
Stage structure is closely related
Sometimes biological stage matters more than chronological age. A population might be divided into eggs, larvae, juveniles and adults, or seedlings and mature plants.
These are stage-structured models. Their projection matrices can include transitions that remain within a stage or move between stages, so they are often more general than the strict Leslie age-class structure.
Main assumptions of a simple Leslie model
The standard formulation assumes that fertility and survival rates are known and remain constant through time, individuals within the same age class are equivalent, and the chosen time step matches the ageing structure.
Density dependence, environmental variability, migration and stochastic demographic events are absent unless explicitly added.
Deterministic and stochastic structured populations
A Leslie matrix usually describes expected numbers deterministically. Real survival and reproduction are random. Stochastic matrix population models can allow survival, fertility or environmental conditions to vary randomly through time.
For small populations, demographic stochasticity can also matter because the number of survivors and offspring is discrete and random.