The Leslie Matrix
Consider a population divided into $m$ age classes. Let $n_x(t)$ be the number of individuals in age class $x$ at time $t$, and form the population vector $\mathbf{n}(t) = (n_1, n_2, \ldots, n_m)^\top$.
The Leslie matrix $\mathbf{L}$ is:
\[\mathbf{L} = \begin{pmatrix} F_1 & F_2 & F_3 & \cdots & F_m \\ P_1 & 0 & 0 & \cdots & 0 \\ 0 & P_2 & 0 & \cdots & 0 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & P_{m-1} & 0 \end{pmatrix}\]where:
- $F_x$ = fecundity of age class $x$ (expected offspring per individual, often $F_x = \ell_x m_x$ with survival to age $x$ built in)
- $P_x = l_{x+1}/l_x$ = probability of surviving from age $x$ to $x+1$ (subdiagonal entries)
The population projects forward as:
\[\mathbf{n}(t+1) = \mathbf{L}\,\mathbf{n}(t) \implies \mathbf{n}(t) = \mathbf{L}^t\,\mathbf{n}(0)\]Example: A three-age-class population with $P_1 = 0.6$, $P_2 = 0.7$, $F_2 = 1.2$, $F_3 = 2.5$:
\[\mathbf{L} = \begin{pmatrix} 0 & 1.2 & 2.5 \\ 0.6 & 0 & 0 \\ 0 & 0.7 & 0 \end{pmatrix}\]Dominant Eigenvalue and Asymptotic Growth
| The characteristic polynomial of $\mathbf{L}$ is obtained from $\det(\mathbf{L} - \lambda\mathbf{I}) = 0$. By the Perron-Frobenius theorem (applied to nonnegative, primitive matrices), $\mathbf{L}$ has a unique dominant real eigenvalue $\lambda_1 > | \lambda_i | $ for all $i \neq 1$. |
The long-run behavior:
\[\mathbf{n}(t) \approx c_1 \lambda_1^t \mathbf{v}_1\]where $\mathbf{v}_1$ is the right eigenvector corresponding to $\lambda_1$ and $c_1$ depends on initial conditions. Thus:
- $\lambda_1 > 1$: population growing
- $\lambda_1 = 1$: stationary population
- $\lambda_1 < 1$: population declining
The finite rate of increase is $\lambda_1$; the continuous-time equivalent is $r = \ln \lambda_1$.
For the Euler-Lotka equation (see below), $\lambda_1$ also satisfies:
\[\sum_{x=1}^{m} F_x \prod_{j=1}^{x-1} P_j \cdot \lambda_1^{-x} = 1\]This provides a scalar equation for $\lambda_1$ when the matrix structure is known.
Stable Age Distribution and Reproductive Value
Stable age distribution: The right eigenvector $\mathbf{v}_1$ gives the proportional age structure approached asymptotically. Normalizing so entries sum to 1:
\[\mathbf{w} = \frac{\mathbf{v}_1}{\mathbf{1}^\top \mathbf{v}_1}\]$w_x$ is the fraction of the population in age class $x$ at the stable age distribution (SAD).
Reproductive value: The left eigenvector $\mathbf{u}_1$ (satisfying $\mathbf{u}_1^\top \mathbf{L} = \lambda_1 \mathbf{u}_1^\top$) gives the reproductive value of each age class — the expected future contribution to population growth, discounted by $\lambda_1$. Normalized so $\mathbf{u}_1^\top \mathbf{w} = 1$:
\[V_x = u_{1,x} \propto \sum_{y=x}^{m} \frac{\ell_y}{\ell_x} m_y \lambda_1^{-(y-x)}\]Reproductive value is maximized at the age of peak reproduction and declines thereafter. In conservation biology, reproductive value weights the importance of preserving different cohorts: losing pre-reproductive individuals is more costly (per capita) than losing post-reproductive ones.
Sensitivity and Elasticity Analysis
Sensitivity of $\lambda_1$ to matrix element $a_{ij}$:
\[s_{ij} = \frac{\partial \lambda_1}{\partial a_{ij}} = \frac{u_i v_j}{\langle \mathbf{u}, \mathbf{v} \rangle}\]where $\mathbf{v} = \mathbf{v}_1$ and $\mathbf{u} = \mathbf{u}_1$ (left eigenvector), and $\langle \mathbf{u}, \mathbf{v}\rangle = \mathbf{u}^\top \mathbf{v}$.
Sensitivity measures the absolute response of $\lambda_1$ to a small change in $a_{ij}$. It is most useful when entries are on comparable scales.
Elasticity (proportional sensitivity) is more ecologically interpretable since it measures the proportional change in $\lambda_1$ for a proportional change in $a_{ij}$:
\[e_{ij} = \frac{a_{ij}}{\lambda_1}\,\frac{\partial \lambda_1}{\partial a_{ij}} = \frac{a_{ij}\,s_{ij}}{\lambda_1}\]A key property: elasticities sum to 1 ($\sum_{ij} e_{ij} = 1$). This allows direct comparison of the relative importance of survival vs fecundity entries.
Conservation insight: For long-lived organisms (sea turtles, whales), $\lambda_1$ is far more sensitive to adult survival elasticities than to juvenile survival or fecundity. Protecting adult females is therefore more effective than protecting nests — a result that caused a major shift in sea turtle conservation strategy (Crouse et al. 1987).
| Life history type | Highest elasticity |
|---|---|
| Long-lived (whales, turtles) | Adult survival $P_{m-1}$ |
| Annual plants | Fecundity $F_1$ |
| Iteroparous vertebrates | Juvenile survival $P_1$ |
Euler-Lotka Equation
The continuous-time analog of the Leslie matrix is the Euler-Lotka equation, which implicitly defines the intrinsic rate of increase $r$ given a survivorship schedule $\ell(a)$ and fecundity schedule $m(a)$:
\[\int_0^\infty e^{-ra}\ell(a)m(a)\,da = 1\]This equation says: the sum of discounted reproductive output across all ages equals 1 at the stable growth rate. The net reproductive rate $R_0 = \int_0^\infty \ell(a)m(a)\,da$ gives the expected lifetime offspring per individual. When $R_0 = 1$, $r = 0$; when $R_0 > 1$, $r > 0$.
Demographic quantities from the Euler-Lotka framework:
- Generation time: $T = \ln R_0 / r \approx \frac{\int_0^\infty a\,\ell(a)m(a)\,da}{R_0}$
-
Damping ratio: $\rho = \lambda_1/\lambda_2 $ (speed of convergence to SAD)
McKendrick-von Foerster Equation
The continuous age-structured PDE (McKendrick 1926, von Foerster 1959) tracks age density $n(a, t)$:
\[\frac{\partial n}{\partial t} + \frac{\partial n}{\partial a} = -\mu(a)\,n(a,t)\]with boundary condition (recruitment):
\[n(0, t) = \int_0^\infty m(a)\,n(a,t)\,da\]and initial condition $n(a, 0) = n_0(a)$. Here $\mu(a)$ is the age-specific mortality rate with $\ell(a) = \exp!\left(-\int_0^a \mu(s)\,ds\right)$.
The steady-state solution with $n(a,t) = c\,e^{rt}\ell(a)$ recovers the Euler-Lotka equation. The PDE is solved numerically using method-of-characteristics or finite difference schemes, and is the foundation of cohort-component demographic projections used by national statistics offices.