Stage-structured populations
Individuals of the same species are not always biologically equivalent. A seed, seedling and mature plant may have completely different survival and reproductive behaviour even when chronological age is not the most useful way to distinguish them.
Why use stage rather than age?
Age tells us how long an organism has lived. Stage tells us its biological condition. Depending on the organism, stage may be determined by development, body size, maturity or reproductive status.
For many plants, insects, amphibians and marine organisms, individuals of the same age can differ greatly in size or developmental state. Their future survival and reproduction may therefore depend more strongly on stage than age.
| Age structure | Stage structure |
|---|---|
| classes determined mainly by chronological age | classes determined by biological state |
| individuals normally advance as they age | individuals may remain in the same stage for several time steps |
| Leslie matrix is a standard form | Lefkovitch or general projection matrices are commonly used |
A simple three-stage population
Consider a plant population divided into seedlings, juveniles and reproductive adults.
The population vector
Let \(S_t\), \(J_t\) and \(A_t\) be the numbers of seedlings, juveniles and adults at time \(t\). The population is represented by
\[\mathbf n_t=\begin{pmatrix}S_t\\J_t\\A_t\end{pmatrix}.\]The total population is \(S_t+J_t+A_t\), but keeping the three components separately preserves information about its biological composition.
What can happen during one time step?
A seedling may survive but remain a seedling, or survive and develop into a juvenile. A juvenile may remain a juvenile or progress to adulthood. An adult may survive and remain an adult. Adults may also produce new seedlings.
These processes can be represented by probabilities or expected contributions.
| Symbol | Biological meaning |
|---|---|
| \(p_S\) | probability a seedling survives and remains a seedling |
| \(g_S\) | probability a seedling survives and grows into the juvenile stage |
| \(p_J\) | probability a juvenile survives and remains juvenile |
| \(g_J\) | probability a juvenile survives and becomes an adult |
| \(p_A\) | probability an adult survives and remains adult |
| \(F\) | expected new seedlings contributed per adult during one time step |
Write each next-stage equation first
The next number of seedlings receives contributions from seedlings that remain seedlings and offspring produced by adults:
\[S_{t+1}=p_SS_t+FA_t.\]The next juvenile population receives juveniles that remain juveniles and seedlings that develop:
\[J_{t+1}=g_SS_t+p_JJ_t.\]The next adult population receives surviving adults and juveniles that develop into adults:
\[A_{t+1}=g_JJ_t+p_AA_t.\]Matrix representation
Putting the three equations together gives
\[\boxed{\mathbf n_{t+1}=P\mathbf n_t},\]with projection matrix
\[\boxed{P=\begin{pmatrix}p_S&0&F\\g_S&p_J&0\\0&g_J&p_A\end{pmatrix}}.\]Why stage matrices differ from Leslie matrices
In a strict age-class Leslie model, an individual normally advances to the next age class after one time step. In a stage model an organism may remain in its current stage, so the diagonal entries can be positive.
This apparently small difference is biologically important. A juvenile tree, for example, may remain juvenile for many years rather than becoming an adult after exactly one year.
A numerical example
Suppose the annual projection matrix is
\[P=\begin{pmatrix}0.20&0&1.50\\0.50&0.60&0\\0&0.25&0.80\end{pmatrix},\qquad\mathbf n_t=\begin{pmatrix}100\\60\\40\end{pmatrix}.\]For seedlings,
\[S_{t+1}=0.20(100)+1.50(40)=80.\]For juveniles,
\[J_{t+1}=0.50(100)+0.60(60)=86.\]For adults,
\[A_{t+1}=0.25(60)+0.80(40)=47.\]Therefore
\[\boxed{\mathbf n_{t+1}=\begin{pmatrix}80\\86\\47\end{pmatrix}}.\]The total changes from 200 to 213, but the more informative result is that the composition of the population has also changed.
Projecting the population through time
Applying the same matrix repeatedly gives
\[\mathbf n_{t+2}=P^2\mathbf n_t,\qquad \mathbf n_{t+k}=P^k\mathbf n_t.\]Long-term population growth
As with age-structured models, the dominant eigenvalue \(\lambda\) of the projection matrix determines the asymptotic multiplication factor under suitable conditions.
| Value | Long-term interpretation |
|---|---|
| \(\lambda>1\) | the population tends to grow |
| \(\lambda=1\) | the population tends to remain constant in overall scale |
| \(\lambda<1\) | the population tends to decline |
Stable stage distribution
The right eigenvector associated with the dominant eigenvalue describes the stable relative distribution among stages. After transients have faded, the proportions of seedlings, juveniles and adults can approach fixed values even while the total population continues to change.
Transient behaviour still matters
The long-term growth factor does not completely describe short-term dynamics. A population with very few reproductive adults may initially decline even if its eventual growth factor exceeds one. Another initial stage distribution may produce temporary rapid growth.
For conservation or management over the next few years, the current stage composition can therefore be as important as the long-term eigenvalue.
Which stage matters most?
Sensitivity analysis asks how the dominant eigenvalue changes when a matrix entry changes. Elasticity gives a proportional version of this comparison.
This can reveal whether long-term growth is particularly sensitive to seedling establishment, juvenile development, adult survival or fertility.
Stage-specific harvesting
Harvesting or mortality can act differently on stages. Removing reproductive adults may have a different long-term effect from removing the same number of juveniles because adults contribute both through their own survival and through reproduction.
Stage-structured models therefore provide a natural framework for evaluating selective harvesting, culling or conservation interventions.
Density dependence
The simple projection matrix assumes fixed transition and fertility parameters. In real populations, these may depend on population density. Crowding might reduce seedling establishment, slow development or reduce fertility.
The matrix then becomes dependent on the current population state, producing a nonlinear stage-structured model.
Environmental and demographic randomness
Survival, development and reproduction are not perfectly predictable. Environmental conditions can make the projection matrix vary between years, while demographic stochasticity makes the actual number of survivors and offspring random, particularly in small populations.
Stochastic stage-structured models can therefore be used to estimate quantities such as extinction probability and the probability of falling below a critical abundance.
Choosing the stages
A good stage classification should reflect important biological differences without creating unnecessary complexity. Useful questions include: Does survival change sharply with size? Does reproduction begin at a particular developmental state? Can individuals remain in the same stage? Does management act differently on different stages?
Where stage-structured models are useful
They are widely applicable to plant populations, insects with developmental stages, amphibians, marine organisms, wildlife conservation and any population in which biological state affects survival, development or reproduction.
From biological life cycle to mathematics
The most reliable way to construct a stage model is to begin with the life cycle rather than with a matrix. Identify the stages, draw every biologically possible contribution between successive time steps, assign each arrow a parameter, write the update equation for each receiving stage, and only then place those equations into matrix form.