Age-structured populations: the Leslie matrix, stable age distribution, and reproductive value

Intuition Beginner

Real populations are not collections of identical individuals. A newborn, a teenager, and a 70-year-old have very different chances of surviving the next year and very different likelihoods of having children. Treating every individual the same ignores the most important source of variation in population growth.

A life table organises this information. For each age class it records two things: what fraction of individuals survive from birth to that age ($l_x$, called survivorship), and how many offspring an individual of that age produces on average ($m_x$, called fertility). Together, these columns tell you the full demographic story of the species.

A human life table shows very low mortality in childhood, a bump in teenage years, and rising mortality after 60 — paired with a fertility schedule that peaks in the mid-20s. A sea turtle life table shows extremely high mortality in the egg and hatchling stages, very low mortality for decades, and then reproduction beginning only at age 20-30.

The Leslie matrix turns the life table into a single mathematical object — a matrix — that projects the entire population forward one time step. You multiply the matrix by the current age-structure vector (how many individuals are in each age class) and get the age structure at the next time step.

No matter what the starting age distribution looks like, the population eventually settles into a stable age distribution where the proportion in each age class stays constant. Once this happens, the whole population grows (or shrinks) by a single rate $\lambda$ — the finite rate of increase — which is the dominant eigenvalue of the Leslie matrix. If $\lambda >1$ the population grows; if $\lambda <1$ it declines.

Different age classes contribute differently to future population growth. A young individual who has not yet reproduced but has many reproductive years ahead has high reproductive value. An individual past reproductive age has reproductive value of zero, even if they survive for many more years. This concept, introduced by R. A. Fisher (1930), answers the question: if you could add one individual to any age class, which would produce the largest long-term increase in population size?

Visual Beginner

Imagine a population organised as a stack of horizontal bars — an age pyramid. Each bar represents one age class, with bar width proportional to the number of individuals. In a growing population, the base is wide (many young) and the top is narrow (few old). In a declining population, the base is narrow and the top is relatively wide.

The Leslie matrix sits beside the pyramid, with arrows showing how individuals flow from one age class to the next (survival, subdiagonal entries) and how newborns enter the first age class from parents of all reproductive ages (fertility, top row).

Worked example Beginner

Consider a plant species with three age classes: seeds (age 1), juveniles (age 2), and adults (age 3). The life table gives:

• Age 1: survivorship $l_1=1.0$, fertility $m_1=0$ (seeds produce no offspring)
• Age 2: survivorship $l_2=0.4$, fertility $m_2=2$ (each juvenile produces 2 seeds on average)
• Age 3: survivorship $l_3=0.2$, fertility $m_3=5$ (each adult produces 5 seeds on average)

Step 1. Compute the net reproductive rate $R_0=(1.0\times 0)+(0.4\times 2)+(0.2\times 5)=0+0.8+1.0=1.8$. Since $R_0>1$, each generation is 1.8 times larger than the previous one — the population is growing.

Step 2. Construct the Leslie matrix. The survival probability from age 1 to age 2 is $s_1=l_2/l_1=0.4/1.0=0.4$. The survival probability from age 2 to age 3 is $s_2=l_3/l_2=0.2/0.4=0.5$. The fertilities go in the top row:

$$\mathbf{L}=\left(\begin{array}{ccc}0& 2& 5\\ 0.4& 0& 0\\ 0& 0.5& 0\end{array}\right).$$

Step 3. Project one time step starting with 100 individuals in each age class: ${\mathbf{n}}_0=(100,100,100{)}^T$.

$${\mathbf{n}}_1=\mathbf{L}{\mathbf{n}}_0=\left(\begin{array}{c}0\times 100+2\times 100+5\times 100\\ 0.4\times 100+0+0\\ 0+0.5\times 100+0\end{array}\right)=\left(\begin{array}{c}700\\ 40\\ 50\end{array}\right).$$

The population shifts dramatically: many newborns (700) from adult reproduction, but few surviving juveniles (40) and adults (50). Repeated projection would show the age proportions converging to the stable age distribution and the total population growing at rate $\lambda$.

Check your understanding Beginner

Formal definition Intermediate+

Life table functions

A life table summarises age-specific survival and reproduction. The standard columns are:

• $l_x$: survivorship — the proportion of individuals surviving from birth to the start of age $x$. By convention $l_0=1$.
• $d_x=l_x-l_{x+1}$: number dying during age interval $x$.
• $q_x=d_x/l_x$: age-specific mortality rate — the probability that an individual of age $x$ dies before reaching age $x+1$.
• $\frac{L_x=l_x+l_{x+1}}{2}$: person-years lived during age interval $x$ (trapezoidal approximation for the survival curve).
• $T_x={\sum }_{y=x}^{\omega }{L}_y$: total person-years remaining to individuals of age $x$.
• $e_x=T_x/l_x$: life expectancy at age $x$ — the average number of additional time units an individual of age $x$ is expected to live.
• $m_x$: age-specific fertility — the average number of female offspring produced per female of age $x$ per time unit (female-dominant formulation assumes only females reproduce).

Net reproductive rate and generation time

The net reproductive rate is:

$$R_0=\sum _{x=0}^{\omega }{l}_x{m}_x.$$

$R_0$ is the expected number of daughters a newborn female produces over her lifetime. If $R_0>1$, the population is growing; if $R_0<1$, it is declining. $R_0$ measures growth per generation, not per time step.

The generation time (mean age of reproduction) is:

$$T=\frac{{\sum }_{x=0}^{\omega }x{l}_x{m}_x}{{\sum }_{x=0}^{\omega }{l}_x{m}_x}=\frac{\sum x{l}_x{m}_x}{R_0}.$$

The approximate relationship between the per-generation rate and the per-time-step rate is $r\approx \ln R_0/T$, or equivalently $\lambda \approx R_0^{1/T}$, where $\lambda =e^r$ is the finite rate of increase.

The Leslie matrix

Leslie (1945) introduced a discrete-time, age-structured population model. The population at time $t$ is represented as a vector ${\mathbf{n}}_t=(n_{0,t},n_{1,t},\dots ,n_{\omega -1,t}{)}^T$ where $n_{i,t}$ is the number of individuals in age class $i$ at time $t$. The population is projected forward by:

$${\mathbf{n}}_{t+1}=\mathbf{L}{\mathbf{n}}_t,$$

where $\mathbf{L}$ is the Leslie matrix:

$$\mathbf{L}=\left(\begin{array}{cccccc}{f}_0& f_1& f_2& \cdots & f_{\omega -2}& f_{\omega -1}\\ s_0& 0& 0& \cdots & 0& 0\\ 0& s_1& 0& \cdots & 0& 0\\ ⋮& & \ddots & & & ⋮\\ 0& \cdots & 0& s_{\omega -2}& 0& \end{array}\right).$$

Here $f_i$ is the fertility of age class $i$ (number of offspring produced per individual of age $i$ that survive to be counted at $t+1$) and $s_i=l_{i+1}/l_i$ is the survival probability from age class $i$ to age class $i+1$. The top row captures births; the subdiagonal captures survival. All other entries are zero because individuals cannot skip age classes.

Dominant eigenvalue and the finite rate of increase

By the Perron-Frobenius theorem, if $\mathbf{L}$ is primitive (some power ${\mathbf{L}}^k$ has all positive entries, which holds when at least two adjacent fertilities $f_i,f_{i+1}$ are positive), then:

1. $\mathbf{L}$ has a unique dominant eigenvalue ${\lambda }_1>0$ that is real and simple.
2. All other eigenvalues satisfy $|{\lambda }_j|<{\lambda }_1$ for $j\ge 2$.
3. ${\lambda }_1$ has a positive right eigenvector $\mathbf{w}$ and a positive left eigenvector $\mathbf{v}$.

The dominant eigenvalue ${\lambda }_1$ is the asymptotic finite rate of increase: as $t\to \infty$, the total population satisfies $N_{t+1}/N_t\to {\lambda }_1$. The intrinsic rate of increase is $r=\ln {\lambda }_1$.

Stable age distribution

The stable age distribution is the right eigenvector $\mathbf{w}$ of ${\lambda }_1$:

$$\mathbf{Lw}={\lambda }_1\mathbf{w},w_i>0.$$

Starting from any positive initial age structure, the population converges to this distribution: ${\mathbf{n}}_t/N_t\to \mathbf{w}/({\sum }_i{w}_i)$ as $t\to \infty$. The proportions $w_i/(\sum w_j)$ give the fraction of the population in each age class at the stable distribution. Once attained, the age structure is self-reproducing: each time step multiplies every age class by ${\lambda }_1$ and shifts survivors to the next class, returning the same proportional structure.

Reproductive value

The reproductive value $\mathbf{v}$ is the left eigenvector of ${\lambda }_1$:

$${\mathbf{v}}^T\mathbf{L}={\lambda }_1{\mathbf{v}}^T,v_i>0.$$

Normalised so that $v_0=1$, the entry $v_i$ measures the expected future reproductive contribution of an individual of age $i$, discounted by the population growth rate. Formally:

$$v_i=\frac{1}{{\lambda }_1^i{l}_i}\sum _{j=i}^{\omega -1}{\lambda }_1^{-(j-i)}{l}_j{m}_j.$$

Reproductive value typically increases from birth to the age of first reproduction (because the individual has survived the risky juvenile period and is about to start reproducing), peaks around the age of maximum reproductive output, and then declines to zero at post-reproductive ages. Fisher (1930) introduced reproductive value as the demographic weight that an individual of each age contributes to the gene pool of the future population.

The reproductive value vector satisfies a useful conservation property: the total reproductive value ${\mathbf{v}}^T{\mathbf{n}}_t$ grows exactly by ${\lambda }_1$ at every time step, even before the population has reached the stable age distribution:

$${\mathbf{v}}^T{\mathbf{n}}_{t+1}={\mathbf{v}}^T\mathbf{L}{\mathbf{n}}_t={\lambda }_1{\mathbf{v}}^T{\mathbf{n}}_t.$$

This makes the total reproductive value a more reliable measure of population status than total population size during transient phases.

Sensitivity and elasticity

Sensitivity measures how much ${\lambda }_1$ changes per unit change in a matrix entry:

$$s_{ij}=\frac{\partial {\lambda }_1}{\partial a_{ij}}=\frac{v_i{w}_j}{{\mathbf{v}}^T\mathbf{w}},$$

where $a_{ij}$ is the $(i,j)$ entry of $\mathbf{L}$, $\mathbf{v}$ is the reproductive value vector, and $\mathbf{w}$ is the stable age distribution vector. The sensitivity $s_{ij}$ gives the absolute change in ${\lambda }_1$ for a small absolute change in $a_{ij}$.

Elasticity is the proportional version:

$$e_{ij}=\frac{a_{ij}}{{\lambda }_1}\frac{\partial {\lambda }_1}{\partial a_{ij}}=\frac{a_{ij}{s}_{ij}}{{\lambda }_1}=\frac{a_{ij}{v}_i{w}_j}{{\lambda }_1{\mathbf{v}}^T\mathbf{w}}.$$

Elasticity $e_{ij}$ gives the proportional change in ${\lambda }_1$ for a proportional change in $a_{ij}$. Elasticities sum to 1 across all matrix entries (${\sum }_{i,j}{e}_{ij}=1$), decomposing the population growth rate into contributions from each demographic process. This decomposition identifies the life stage where management intervention has the greatest effect on population growth.

The Euler-Lotka equation

For continuous-time age-structured populations, the relationship between the life table and the population growth rate $r$ is given by the Euler-Lotka equation:

$${\int }_0^{\infty }{e}^{-ra}l(a)m(a)da=1,$$

where $l(a)$ is the continuous survivorship function and $m(a)$ is the continuous fertility function. This is an implicit equation for $r$ with a unique real root when $R_0>0$. For $r=0$, the left side equals $R_0$; the root $r>0$ if and only if $R_0>1$. In the discrete case with age classes of width $\Delta$, the Euler-Lotka equation becomes:

$$\sum _{x=0}^{\omega }{e}^{-rx}{l}_x{m}_x=1,$$

and the connection to the Leslie matrix is $r=\ln {\lambda }_1$. The Euler-Lotka equation is the continuous-time analogue of the characteristic equation $\det (\mathbf{L}-\lambda \mathbf{I})=0$.

Counterexamples to common slips

• $R_0$ and $\lambda$ measure different things. $R_0$ is the per-generation multiplication factor; $\lambda$ is the per-time-step factor. A species with $R_0=2$ and $T=25$ years (elephants) has $\lambda \approx 2^{1/25}\approx 1.028$, growing at 2.8% per year — slow despite a generation doubling. A species with $R_0=1.5$ and $T=1$ year (annual plant) has $\lambda \approx 1.5$, growing at 50% per year. The generation time mediates the relationship.
• The Leslie matrix assumes synchronous, discrete time steps. All individuals advance one age class per time step regardless of when during the interval they were born. This approximation is poor for species with continuous breeding and is corrected by birth-flow formulations where survival and fertility are integrated over the interval.
• The stable age distribution is not the same as the observed age distribution. Many real populations are far from their stable distribution due to recent disturbances, seasonal breeding, or harvesting. The transient dynamics during convergence can last many generations, especially for long-lived species with low damping ratios.

Key theorem with proof Intermediate+

Theorem (Asymptotic growth of Leslie matrix populations). Let $\mathbf{L}$ be a primitive Leslie matrix with dominant eigenvalue ${\lambda }_1$, right eigenvector $\mathbf{w}$, and left eigenvector $\mathbf{v}$, normalised so that ${\mathbf{v}}^T\mathbf{w}=1$. For any initial population vector ${\mathbf{n}}_0>0$:

$$\underset{t\to \infty }{\lim }\frac{{\mathbf{n}}_t}{{\lambda }_1^t}=\mathbf{w}\cdot ({\mathbf{v}}^T{\mathbf{n}}_0).$$

In particular, ${\mathbf{n}}_t/N_t\to \mathbf{w}/(1^T\mathbf{w})$ (convergence to the stable age distribution) and $N_{t+1}/N_t\to {\lambda }_1$ (convergence to the asymptotic growth rate).

Proof. Since $\mathbf{L}$ is primitive, by the Perron-Frobenius theorem it has a dominant eigenvalue ${\lambda }_1>0$ with $|{\lambda }_j|<{\lambda }_1$ for all $j\ge 2$. The eigenvectors $\{\mathbf{w},\mathbf{v}\}$ form part of a basis for ${\mathbb{R}}^{\omega }$. Decompose the initial vector:

$${\mathbf{n}}_0=c_1\mathbf{w}+\sum _{j=2}^{\omega }{c}_j{\mathbf{u}}_j,$$

where ${\mathbf{u}}_j$ are generalised eigenvectors corresponding to ${\lambda }_j$ and $c_1={\mathbf{v}}^T{\mathbf{n}}_0/({\mathbf{v}}^T\mathbf{w})$. Applying ${\mathbf{L}}^t$:

$${\mathbf{n}}_t={\mathbf{L}}^t{\mathbf{n}}_0=c_1{\lambda }_1^t\mathbf{w}+\sum _{j=2}^{\omega }{c}_j{\lambda }_j^t{\mathbf{u}}_j.$$

Dividing by ${\lambda }_1^t$:

$$\frac{{\mathbf{n}}_t}{{\lambda }_1^t}=c_1\mathbf{w}+\sum _{j=2}^{\omega }{c}_j{\left(\frac{{\lambda }_j}{{\lambda }_1}\right)}^t{\mathbf{u}}_j.$$

Since $|{\lambda }_j/{\lambda }_1|<1$ for all $j\ge 2$, each term in the sum converges to zero as $t\to \infty$. Therefore:

$$\underset{t\to \infty }{\lim }\frac{{\mathbf{n}}_t}{{\lambda }_1^t}=c_1\mathbf{w}=\mathbf{w}\cdot \frac{{\mathbf{v}}^T{\mathbf{n}}_0}{{\mathbf{v}}^T\mathbf{w}}.$$

Normalising ${\mathbf{v}}^T\mathbf{w}=1$ gives the stated result. Convergence to the stable age distribution follows by dividing by $N_t=1^T{\mathbf{n}}_t$ and noting that ${\mathbf{n}}_t/N_t\to c_1{\lambda }_1^t\mathbf{w}/(c_1{\lambda }_1^t{1}^T\mathbf{w})=\mathbf{w}/(1^T\mathbf{w})$. $\square$

Bridge. This proof is a direct application of the Perron-Frobenius theorem for non-negative matrices, which also underpins the metapopulation capacity result in 19.09.02 pending (where the landscape matrix's dominant eigenvalue determines persistence). The convergence rate is governed by the damping ratio $\rho ={\lambda }_1/|{\lambda }_2|$ — the ratio of the dominant to the subdominant eigenvalue. A damping ratio close to 1 means slow convergence (transients last many time steps); a large damping ratio means rapid convergence. The role of the damping ratio parallels the ratio $r_{\text{eff}}/K_{\text{eff}}$ in the logistic convergence of 19.09.01: both control how quickly a population settles to its asymptotic behaviour.

Exercises Intermediate+

Stage-based and continuous population models Master

Lefkovitch stage-based models

Many organisms are difficult to age but easy to classify by developmental stage (e.g., seed, seedling, sapling, canopy tree for plants; egg, larva, pupa, adult for insects). Lefkovitch (1965, Bull. Entomol. Soc. Am. 11, 190-195) generalised the Leslie matrix to stage-based projection matrices where individuals can remain in the same stage or skip stages. The Lefkovitch matrix $\mathbf{M}$ has the form:

$$M_{ij}=\{\begin{array}{ll}{\sigma }_i{G}_i& \text{(transition to next stage)}\\ {\sigma }_i(1-G_i)& \text{(stasis in current stage, diagonal)}\\ F_i& \text{(fertility, top row entries)}.\end{array}$$

Here ${\sigma }_i$ is the stage-specific survival probability and $G_i$ is the probability of graduating to the next stage (given survival). Unlike the Leslie matrix, the Lefkovitch matrix allows positive entries on the diagonal (individuals that survive but do not advance) and possibly above the diagonal (individuals that regress to earlier stages, e.g., size reduction in plants after herbivory). The same Perron-Frobenius theory applies: a primitive Lefkovitch matrix has a dominant eigenvalue ${\lambda }_1$ giving the population growth rate, with right and left eigenvectors giving the stable stage distribution and reproductive value by stage.

Stage-based models are preferred over age-based models for organisms where demographic rates depend more on size or developmental stage than on chronological age. Trees, for example, are classified by diameter class rather than age because growth rate varies enormously among individuals of the same age depending on light, soil, and competition.

Usher matrix

The Usher matrix (Usher 1966, J. Ecol. 54, 99-108) is a special case of the Lefkovitch matrix used primarily in forestry. It classifies trees by diameter classes and allows stasis (remaining in the same diameter class) as well as progression. The matrix has the same structure as the Leslie matrix but with additional positive entries on the diagonal:

$$\mathbf{U}=\left(\begin{array}{ccccc}{f}_1+p_1& f_2& f_3& \cdots & f_{\omega }\\ g_1& p_2& 0& \cdots & 0\\ 0& g_2& p_3& \cdots & 0\\ ⋮& & \ddots & \ddots & ⋮\\ 0& \cdots & 0& g_{\omega -1}& p_{\omega }\end{array}\right),$$

where $p_i$ is the probability of surviving and staying in class $i$, and $g_i$ is the probability of surviving and growing into the next class. The Usher model is standard in yield-per-recruit analysis in fisheries and forestry.

Integral projection models (IPMs)

Matrix models require discretising the population into discrete classes, but many traits (body size, weight) vary continuously. Easterling et al. (2000, Ecology 81, 694-708) introduced integral projection models (IPMs) that replace the matrix equation with an integral kernel:

$$n(y,t+1)={\int }_L^{U}k(y,x)n(x,t)dx,$$

where $n(x,t)$ is the density of individuals of size $x$ at time $t$, and $k(y,x)$ is the kernel giving the number of offspring of size $y$ produced per individual of size $x$ (incorporating both survival-growth and reproduction). The kernel is typically decomposed as:

$$k(y,x)=\underset{\text{survival and growth}}{\underbrace{s(x)g(y,x)}}+\underset{\text{fertility and offspring size distribution}}{\underbrace{f(x)f_d(y)}},$$

where $s(x)$ is the survival function, $g(y,x)$ is the probability density of growing from size $x$ to size $y$ given survival, $f(x)$ is the fertility function, and $f_d(y)$ is the probability density of offspring size.

IPMs are estimated from regression models linking survival, growth, and fertility to individual size. The dominant eigenvalue of the kernel operator gives ${\lambda }_1$; the right eigenfunction gives the stable size distribution; and sensitivity and elasticity can be computed analogously to matrix models. IPMs have become the standard tool for plant population biology because they avoid arbitrary size-class boundaries and capture continuous variation in demographic rates.

Stochastic matrix models

Environmental stochasticity — temporal variation in demographic rates driven by weather, resource availability, or disturbance — is incorporated by allowing the projection matrix to vary from year to year. In a stochastic matrix model, the matrix at time $t$ is drawn from a distribution $\{{\mathbf{L}}_1,\dots ,{\mathbf{L}}_k\}$ according to an environmental state:

$${\mathbf{n}}_{t+1}={\mathbf{L}}_t{\mathbf{n}}_t,{\mathbf{L}}_t\in \{{\mathbf{L}}_1,\dots ,{\mathbf{L}}_k\}.$$

The stochastic growth rate is:

$$\log {\lambda }_S=\underset{t\to \infty }{\lim }\frac{1}{t}\log N_t=\underset{t\to \infty }{\lim }\frac{1}{t}\sum _{\tau =0}^{t-1}\log {\lambda }_1({\mathbf{L}}_{\tau }),$$

where ${\lambda }_1({\mathbf{L}}_{\tau })$ is the dominant eigenvalue of the matrix realised at time $\tau$. By Jensen's inequality, $\log {\lambda }_S\le E[\log {\lambda }_1(\mathbf{L})]\le \log E[{\lambda }_1(\mathbf{L})]=\log \bar{\lambda }$: environmental variation always reduces the long-run growth rate below the deterministic average. The difference $\log \bar{\lambda }-\log {\lambda }_S$ is the small-noise approximation of the environmental variance effect, proportional to the variance in $\log {\lambda }_1$.

Demographic stochasticity — random variation in individual fates (birth or death, producing 0 or 3 offspring) — operates even in constant environments. In a finite population of size $N$, demographic stochasticity introduces a variance term ${\sigma }_d^2/N$ in the per-capita growth rate. As $N\to \infty$, demographic stochasticity vanishes; at small $N$, it can drive populations extinct even when ${\lambda }_S>1$. The minimum viable population (MVP) is the smallest $N$ at which the extinction probability over a specified time horizon is acceptably low, given both environmental and demographic stochasticity.

Transient dynamics and damping ratio

The asymptotic growth rate ${\lambda }_1$ describes the population only after the stable age distribution is attained. Before that, transient dynamics dominate, and the population may grow faster or slower than ${\lambda }_1$ predicts, or even decline while ${\lambda }_1>1$.

The damping ratio $\rho ={\lambda }_1/|{\lambda }_2|$ measures the rate of convergence to the stable distribution. The time to converge within a fraction $ϵ$ of the stable distribution is approximately:

$$t_{ϵ}\approx \frac{\log (1/ϵ)}{\log \rho }.$$

For loggerhead sea turtles, the damping ratio is approximately 1.5, giving $t_{ϵ}\approx 50$ years for $ϵ=0.01$ — the transient phase lasts decades, which is why short-term population projections for turtles can be misleading.

Population momentum (Keyfitz 1971, Demography 8, 71-80) occurs when a population that has reached replacement fertility (${\lambda }_1=1$) continues to grow because the age distribution is younger than the stable distribution. The large cohort of young individuals must pass through their reproductive years before growth ceases. Human population momentum is estimated at 1.5-2 billion additional people even if every country immediately reached replacement fertility, because the current young age structure in developing nations guarantees decades of continued growth.

Sensitivity analysis and applications Master

Life table response experiments (LTRE)

Caswell (1989, Ecology 70, 1654-1664) developed life table response experiments (LTRE) as a method to decompose differences in ${\lambda }_1$ between populations (or treatment groups) into contributions from differences in individual matrix entries. For two populations with matrices ${\mathbf{A}}^{(1)}$ and ${\mathbf{A}}^{(2)}$ and growth rates ${\lambda }^{(1)}$ and ${\lambda }^{(2)}$:

$${\lambda }^{(2)}-{\lambda }^{(1)}\approx \sum _{i,j}\left(a_{ij}^{(2)}-a_{ij}^{(1)}\right)s_{ij}^{(\ast )},$$

where $s_{ij}^{(\ast )}$ are sensitivities evaluated at a reference matrix (typically $\bar{\mathbf{A}}=({\mathbf{A}}^{(1)}+{\mathbf{A}}^{(2)})/2$). Each term in the sum gives the contribution of the difference in the $(i,j)$ vital rate to the total difference in $\lambda$. This decomposition identifies which vital rate differences are demographically important — not just statistically significant.

LTRE has been applied to compare populations across environmental gradients, treatment groups in experiments, and species in comparative demography studies. A fixed-design LTRE treats group membership as fixed; a random-design LTRE treats matrices as drawn from a distribution and partitions variance in $\lambda$ into variance components.

Conservation: sea turtle population models

Crouse et al. (1987, Ecology 68, 1412-1423) applied a stage-based matrix model to loggerhead sea turtles (Caretta caretta) and showed that population growth was most sensitive to survival of large juveniles and sub-adults, not to egg or hatchling survival. This result was counterintuitive: conservation efforts had focused on protecting nesting beaches and eggs (the most visible life stage), but elasticity analysis showed that a 10% improvement in juvenile survival increased $\lambda$ far more than a 10% improvement in hatchling survival. The policy implication was that turtle excluder devices (TEDs) on fishing trawls — which reduce juvenile and adult bycatch mortality — are far more effective than beach protection alone. This study is the canonical example of how matrix population models inform conservation prioritisation.

Fisheries: yield-per-recruit analysis

The Beverton-Holt yield-per-recruit model (1957, Fishery Investigations Series II 19, 1-533) is a Leslie-like model that tracks a cohort of fish from recruitment (entry into the fishery) through growth and natural mortality, with fishing mortality applied at each age. The model computes the expected yield per recruit $Y/R$ as a function of age at first capture $t_c$ and fishing mortality $F$:

$$\frac{Y}{R}={\int }_{t_c}^{\infty }FW(t)e^{-(F+M)(t-t_c)}dt,$$

where $W(t)$ is the weight at age $t$ (from a von Bertalanffy growth curve) and $M$ is natural mortality. The maximum $Y/R$ occurs at intermediate $F$ (the maximum sustainable yield in the yield-per-recruit sense), but this economic optimum ignores the spawning stock biomass per recruit $SSB/R$, which declines monotonically with $F$. The $F_{x\%}$ reference point is the fishing mortality at which $SSB/R$ is $x\%$ of its unfished value; $F_{40\%}$ (40% of unfished spawning biomass per recruit) is a common precautionary target.

Human demography: population projections

National population projections use cohort-component projection matrices — essentially large Leslie matrices with single-year or five-year age classes, separate male and female components, and migration. The United Nations Population Division projects global population using a probabilistic version that samples fertility and mortality trajectories from distributions estimated from country-level data. The 2024 revision projects a peak global population of approximately 10.3 billion in the 2080s, driven by declining fertility rates that reduce ${\lambda }_1$ below replacement (${\lambda }_1<1$) in most countries.

The total fertility rate (TFR) — the average number of children a woman would bear if she survived through all reproductive ages — is the human demographic analogue of $R_0$. Replacement fertility is TFR $\approx 2.1$ (slightly above 2 to account for infant mortality and the sex ratio at birth). When TFR drops below 2.1, ${\lambda }_1<1$ and the population eventually declines, but the timing depends on population momentum: countries with young age structures (e.g., many in sub-Saharan Africa) continue growing for decades after TFR falls below replacement.

Lower-level sensitivities

Matrix entries are composite parameters built from lower-level vital rates (survival, growth, reproduction). Lower-level sensitivities propagate through the chain rule:

$$\frac{\partial \lambda }{\partial \theta }=\sum _{i,j}\frac{\partial \lambda }{\partial a_{ij}}\frac{\partial a_{ij}}{\partial \theta }=\sum _{i,j}{s}_{ij}\frac{\partial a_{ij}}{\partial \theta },$$

where $\theta$ is a lower-level parameter (e.g., the probability of surviving a specific cause of mortality). This allows sensitivity analysis at the level of management actions: if a conservation intervention improves adult survival by 5%, the lower-level sensitivity converts this to a change in $\lambda$. Morris and Doak (2002, Quantitative Conservation Biology, Sinauer) developed these methods extensively for conservation applications.

Connections Master

• Population ecology and Lotka-Volterra 19.09.01. The Leslie matrix is the age-structured generalisation of the exponential growth model $dN/dt=rN$. The dominant eigenvalue ${\lambda }_1=e^r$ is the geometric growth rate that appears in the simple exponential model, now derived from age-specific vital rates rather than assumed constant. The logistic model's carrying capacity $K$ can be combined with the Leslie matrix by making survival or fertility density-dependent, producing an age-structured version of logistic growth where the stable age distribution and $K$ are jointly determined.

• Metapopulation dynamics 19.09.02 pending. Metapopulation models treat space explicitly; Leslie models treat age explicitly. Combined spatially structured, age-structured models track age structure within each patch and dispersal between patches. The metapopulation capacity ${\lambda }_M$ from 19.09.02 pending plays the same role as the Leslie matrix eigenvalue ${\lambda }_1$: both are dominant eigenvalues that determine persistence. Source patches have local ${\lambda }_1>1$; sink patches have ${\lambda }_1<1$ but are maintained by immigration, exactly as in the source-sink framework.

• Genetic drift 19.04.01. The effective population size $N_e$ depends on the age structure through variation in reproductive success among age classes. An age-structured population with highly variable reproductive output across ages (e.g., only a few old individuals reproduce) has $N_e\ll N$ because the effective number of breeders is small. The variance effective size for an age-structured population is $N_e\approx 4N/(2+{\sigma }_k^2/\bar{k})$ where ${\sigma }_k^2$ is the variance in lifetime reproductive success and $\bar{k}$ is the mean. Species with high reproductive skew (sea turtles, many plants) have the smallest $N_e/N$ ratios.

• Migration and gene flow 19.02.04 pending. Age structure interacts with gene flow because dispersal is often age-specific. Juveniles disperse more than adults in many bird and mammal species, so the effective migration rate depends on the age distribution of the population. A population skewed toward juveniles (growing population) exports more migrants than one skewed toward adults (stable or declining population), creating an eco-evolutionary feedback where demography shapes gene flow.

• Conservation biology 19.14.01. Matrix population models are the primary quantitative tool in conservation biology. Population viability analysis (PVA) uses stochastic matrix models to estimate extinction probability over specified time horizons. The sensitivity/elasticity framework identifies which vital rates to target with management interventions. The IUCN Red List criteria use population decline rates (directly related to ${\lambda }_1$) to assess extinction risk: a decline of 80% over 10 years or 3 generations (whichever is longer) qualifies a species as Critically Endangered.

Historical & philosophical context Master

Patrick Holt Leslie (1900-1972), a physiologist and statistician at the Bureau of Animal Population at Oxford, published the Leslie matrix in two landmark papers in 1945 (Biometrika 33, 183-212 and 213-231). Leslie developed the model while working under Charles Elton, who had established the Bureau in 1932 to study the cyclic fluctuations of voles, lemmings, and other small mammals in the vicinity of Oxford. The Bureau's long-term data on voles revealed dramatic population cycles, and Leslie sought a mathematical framework that could capture the role of age structure in driving these oscillations. His key insight was to represent the population as a vector and the demographic transitions as a matrix, so that the machinery of linear algebra — eigenvalues, eigenvectors — could be brought to bear on the biological problem ^{[Leslie 1945]}.

Leslie's work built on the life-table tradition in actuarial science and demography. The first life table was constructed by Edmund Halley (1693) for the city of Breslau, using mortality data to compute annuity values. Raymond Pearl (1928) brought life-table methods into biology through laboratory studies of Drosophila populations, and Deevey (1947, Q. Rev. Biol. 22, 283-314) surveyed life tables across species, establishing the survivorship curve types (Type I: low mortality until old age, e.g., humans; Type II: constant mortality rate, e.g., birds; Type III: extremely high juvenile mortality followed by low adult mortality, e.g., sea turtles, oysters) that remain standard in ecology ^{[Deevey 1947]}.

The reproductive value concept was introduced by R. A. Fisher in The Genetical Theory of Natural Selection (1930, Clarendon Press). Fisher was interested in how natural selection acts on organisms of different ages. His key observation was that the "value" of an individual to the future population depends not on its current reproductive output alone but on all future reproduction discounted by the population growth rate. Fisher's reproductive value resolves the puzzle of why selection does not eliminate senescence: mutations that act late in life have low reproductive value and are therefore weakly opposed by selection, allowing the accumulation of deleterious late-acting alleles (the evolutionary theory of ageing, Medawar 1952, An Unsolved Problem of Biology, Lewis; Williams 1957, Evolution 11, 398-411) ^{[Fisher 1930]}.

The mathematical theory of matrix population models was unified by Hal Caswell in Matrix Population Models (1st ed. 1989, 2nd ed. 2001, Sinauer). Caswell's monograph brought together Leslie's age-class models, Lefkovitch's stage-class models, and Usher's size-class models under a single mathematical framework based on the Perron-Frobenius theorem and perturbation theory. Caswell also developed the formal theory of sensitivity and elasticity analysis, showing that the sensitivity formula $\partial \lambda /\partial a_{ij}=v_i{w}_j/{\mathbf{v}}^T\mathbf{w}$ applies to any projection matrix, not just Leslie matrices, and can be extended to lower-level parameters through the chain rule ^{[Caswell 2001]}.

The application of matrix models to conservation biology was catalysed by the sea turtle study of Crouse, Crowder, and Caswell (1987, Ecology 68, 1412-1423). Before this study, conservation efforts for sea turtles focused almost exclusively on protecting nesting beaches and eggs, based on the intuitive reasoning that the hundreds of eggs laid by each female were the most vulnerable life stage. Crouse et al. showed that the elasticity of $\lambda$ to juvenile and sub-adult survival was an order of magnitude larger than the elasticity to egg or hatchling survival, because eggs suffer massive mortality regardless of beach protection while juvenile turtles are killed in large numbers by shrimp trawls. This result led directly to federal regulations requiring turtle excluder devices (TEDs) on US shrimp boats — one of the clearest examples of a mathematical model directly shaping wildlife policy ^{[Crouse et al. 1987]}.

The integral projection model (IPM) was introduced by Easterling, Ellner, and Dixon (2000, Ecology 81, 694-708) as a continuous-size alternative to discrete matrix models. The IPM was motivated by the difficulty of classifying plants into discrete size classes when growth is continuous and the choice of class boundaries is arbitrary. Ellner and Rees (2006, Am. Nat. 167, 410-428) extended the IPM to include stochastic environments, and the model has since become the standard tool for plant demography, replacing matrix models in most applications where size (rather than discrete stage) is the relevant state variable.

Bibliography Master

@article{Leslie1945,
  author = {Leslie, P. H.},
  title = {On the use of matrices in certain population mathematics},
  journal = {Biometrika},
  volume = {33},
  pages = {183--212},
  year = {1945},
}

@book{Fisher1930,
  author = {Fisher, R. A.},
  title = {The Genetical Theory of Natural Selection},
  publisher = {Clarendon Press},
  year = {1930},
}

@article{Deevey1947,
  author = {Deevey, Edward S.},
  title = {Life tables for natural populations of animals},
  journal = {Q. Rev. Biol.},
  volume = {22},
  pages = {283--314},
  year = {1947},
}

@article{Lefkovitch1965,
  author = {Lefkovitch, L. P.},
  title = {The study of population growth in organisms grouped by stages},
  journal = {Biometrics},
  volume = {21},
  pages = {1--18},
  year = {1965},
}

@article{Usher1966,
  author = {Usher, M. B.},
  title = {A matrix approach to the management of renewable resources, with special reference to selection forests},
  journal = {J. Appl. Ecol.},
  volume = {3},
  pages = {355--367},
  year = {1966},
}

@article{Crouse1987,
  author = {Crouse, Deborah T. and Crowder, Larry B. and Caswell, Hal},
  title = {A stage-based population model for loggerhead sea turtles and implications for conservation},
  journal = {Ecology},
  volume = {68},
  pages = {1412--1423},
  year = {1987},
}

@book{Caswell2001,
  author = {Caswell, Hal},
  title = {Matrix Population Models: Construction, Analysis, and Interpretation},
  edition = {2nd},
  publisher = {Sinauer Associates},
  year = {2001},
}

@article{Easterling2000,
  author = {Easterling, Michael R. and Ellner, Stephen P. and Dixon, Philip M.},
  title = {Size-specific sensitivity: applying a new structured population model},
  journal = {Ecology},
  volume = {81},
  pages = {694--708},
  year = {2000},
}

@article{EllnerRees2006,
  author = {Ellner, Stephen P. and Rees, Mark},
  title = {Integral projection models for species with complex demography},
  journal = {Am. Nat.},
  volume = {167},
  pages = {410--428},
  year = {2006},
}

@article{Keyfitz1971,
  author = {Keyfitz, Nathan},
  title = {On the momentum of population growth},
  journal = {Demography},
  volume = {8},
  pages = {71--80},
  year = {1971},
}

@article{Caswell1989,
  author = {Caswell, Hal},
  title = {Analysis of life table response experiments I. Decomposition of effects on population growth rate},
  journal = {Ecology},
  volume = {70},
  pages = {1654--1664},
  year = {1989},
}

@book{BevertonHolt1957,
  author = {Beverton, Raymond J. H. and Holt, Sidney J.},
  title = {On the Dynamics of Exploited Fish Populations},
  publisher = {Her Majesty's Stationery Office},
  year = {1957},
}

@book{MorrisDoak2002,
  author = {Morris, William F. and Doak, Daniel F.},
  title = {Quantitative Conservation Biology: Theory and Practice of Population Viability Analysis},
  publisher = {Sinauer Associates},
  year = {2002},
}

@book{RicklefsRelyea2014,
  author = {Ricklefs, Robert E. and Relyea, Rick},
  title = {Ecology: The Economy of Nature},
  edition = {8th},
  publisher = {W. H. Freeman},
  year = {2014},
}

@book{HartlClark2007,
  author = {Hartl, Daniel L. and Clark, Andrew G.},
  title = {Principles of Population Genetics},
  edition = {4th},
  publisher = {Sinauer},
  year = {2007},
}