McKendrick–von Foerster Equations

Updated 5 February 2026

McKendrick–von Foerster equations are first-order PDEs modeling structured populations by capturing dynamics of aging, death, and renewal through nonlocal boundary conditions.
They combine operator theory, spectral analysis, and measure-theoretic approaches to establish exponential convergence and well-posedness.
Numerical schemes such as finite difference methods and the Escalator Boxcar Train method provide practical tools for simulating these complex population dynamics.

The McKendrick–von Foerster Equations constitute a canonical mathematical framework for the dynamics of structured populations, most typically age or size-structured. Formulated as first-order partial differential equations (PDEs) with nonlocal boundary conditions, these models rigorously encapsulate vital demographic processes—aging, death, renewal (birth), and in extensions, fragmentation or fission. Their well-posedness, spectral properties, measure-theoretic generalizations, numerical analysis, and applications in population biology, epidemiology, and ecology have been the subject of a large and deep literature.

1. Mathematical Formulation and Abstract Cauchy Problem

The classical McKendrick–von Foerster equation describes the density $u(t,a)$ of individuals at age $a \ge 0$ and time $t \ge 0$ , evolving according to

$\begin{cases} \partial_t u(t,a) + \partial_a u(t,a) + B(a) u(t,a) = 0, \quad t>0,\, a>0, \ u(t,0) = \int_0^{\infty} \beta(a)\,u(t,a)\,da, \quad t>0, \ u(0,a) = u_0(a), \quad a > 0. \end{cases}$

Here, $B(a)$ is the death (or exiting) rate at age $a$ , and $\beta(a)$ is the fertility (renewal) kernel. This system is interpreted on the Banach lattice $X = L^1(\mathbb{R}_+)$ , or more generally in a weighted $L^1(w\,da)$ if $B(a)$ is unbounded at infinity.

In operator-theoretic terms, the dynamics are triggered by the abstract Cauchy problem: $\frac{d}{dt} u(t) = (L + A) u(t), \quad u(0) = u_0,$ with \begin{align*} L u &= -\partial_a u - B(a) u, \quad D(L) = {u \in L¹ : \partial_a u \in L^1,\, u(0) = 0}, \ A u(a) &= \delta_{a=0} \int_0^{\infty} \beta(s) u(s) ds. \end{align*} The operator $L$ is hypodissipative and generates a positive semigroup, while $A$ is a bounded, rank-one (hence compact) operator (Mischler et al., 2013).

2. Spectral Theory, Long-Time Asymptotics, and the Spectral Gap

The spectral properties of the semigroup generated by $\Lambda = L + A$ are foundational:

Main Spectral Theorem: Under natural positivity, irreducibility, and regularity conditions on $B$ and $\beta$ , the spectral bound $s(\Lambda)$ is an algebraically simple real eigenvalue $\lambda^* > 0$ . There exist strictly positive eigenvectors $\phi \in D(\Lambda)$ , $\psi \in D(\Lambda^*)$ , normalized such that $\int_0^\infty \psi(a)\phi(a) da = 1$ , $\|\phi\|_{L^1} = 1$ .
Asymptotic Behavior: The solution decomposes as

$u(t, a) = e^{\lambda^* t} \langle \psi, u_0 \rangle \phi(a) + R(t, a), \quad \|R(t, \cdot)\|_{L^1} \leq C' e^{-(\lambda^* - a) t} \|u_0\|_{L^1},$

where $a < \lambda^*$ tracks a uniform spectral gap. Thus, any nontrivial initial datum converges exponentially in $L^1$ to the unique asymptotic profile $\phi$ at the population Malthusian parameter $\lambda^*$ (Mischler et al., 2013).

The proof strategy involves: (i) Duhamel/ Dyson–Phillips expansion expressing the semigroup as a sum of the semigroup generated by $L$ and correction terms, (ii) detailed spectral mapping and Weyl-type theorems, (iii) constructive Krein–Rutman approach for positivity and simple leading eigenvalue, and (iv) explicit spectral gap analysis via compactness of the regularizing component $A$ .

3. Measure-Valued Extensions and Generalized Entropy Methods

Modern treatments rigorously extend the McKendrick–von Foerster framework to the space of bounded nonnegative Radon measures $\mathcal{M}^+([0,\infty))$ , which encompasses arbitrary initial data—including singular (e.g., sum of Dirac masses) and weakly convergent limits—in place of function spaces. In this setting, solutions admit distributional sense: $\partial_t \mu_t + \partial_a \mu_t + B(a) \mu_t = 0; \quad \mu_t(\{0\}) = \int_0^\infty b(a)\,\mu_t(da)$ (Gwiazda et al., 2016).

Key results:

Existence and Uniqueness: For any measure initial datum, the solution exists and is unique.
Exponential Convergence: There exist explicit weights $\varphi\in C_b([0,\infty))$ and constants $\alpha>0$ such that

$\int_0^\infty \varphi(a)\left|\mu_t(da) - m_0 N(a) da\right| \leq e^{-\alpha t}\int_0^\infty \varphi(a)\left|\mu_0(da) - m_0 N(a) da\right|,$

where $N(a)$ is the stationary age density and $m_0 = \int P(a)\mu_0(da)$ the preserved mass.

Technical Approach: The generalized entropy method, extended to measures using Reshetnyak’s theorem and convex-analytic recession functions, ensures the contraction property even in the absence of strong continuity in the total variation norm (Gwiazda et al., 2016).

The McKendrick–von Foerster model generalizes to cover several advanced structured population scenarios:

Size- or Trait-Structured Models: Replacing the structure variable by physiological size or generic trait, and including non-trivial growth $g(x)$ . Growth–fragmentation equations with McKendrick–von Foerster boundary conditions fit in this class. The nonlocal renewal term is then

$g(0) u(t,0) = \int_0^\infty B(x) u(t,x) dx$

(Banasiak et al., 2022).

Two-Sex and Multi-Compartment Systems: Systems of coupled McKendrick–von Foerster equations model interacting populations, e.g., two-sex (Fredrickson–Hoppensteadt) or age-and-sex-structured human populations. Each subpopulation is governed by a coupled PDE with interlinked nonlocal boundary or source terms (Pokojovy et al., 2014, Carrillo et al., 2018).
Stochastic Generalizations: The deterministic PDE emerges as the first-moment closure of a stochastic, age-dependent birth-death process. Fully stochastic kinetic theories yield BBGKY-type hierarchies governing evolution of multiparticle age distributions, and recover the McKendrick–von Foerster equation in the mean-field, population-independent-rate limit. Doi–Peliti field-theoretic and path-integral approaches formalize these connections (Greenman et al., 2015, Chou et al., 2015, Greenman, 2015).
Mutation and Fragmentation: Systems with multiple types or genetic classes, coupled at division by a mutation or fragmentation kernel, admit matrix-valued generalizations. Under singular scaling (rapid cycling), the system reduces in the hydrodynamic limit to a network of coupled ODEs for class sizes (Banasiak et al., 2016).

5. Numerical Schemes and Computational Analysis

Discretization of the McKendrick–von Foerster equation and its extensions is an active area of research:

Finite Difference Methods: Both explicit and implicit finite difference schemes (including Crank–Nicolson θ-schemes) are implemented for the age and time variables, with consistency and convergence rates $O(h+\tau)$ for smooth solutions; proof relies on semigroup theory and stability via resolvent estimates (Pokojovy et al., 2014).
Escalator Boxcar Train (EBT) Method: This cohort-based method approximates the structured density by a finite sum of moving Dirac delta functions (“cohorts”) with ODE-prescribed mass and location evolution. Convergence is established in the space of nonnegative Radon measures equipped with the flat (bounded–Lipschitz) metric. This approach generalizes to coupled systems (e.g., two-sex models) and nonlinear source/boundary conditions (Carrillo et al., 2018, Gwiazda et al., 2015).
Diffusive and Nonlinear Variants: Finite-difference and hybrid characteristic–finite-difference schemes for equations with age diffusion and nonlinear, nonlocal (Robin-type) boundary conditions achieve stability and convergence under suitable CFL-type conditions. Nonlocality and nonlinearity in boundary conditions are addressed via stability-with-thresholds and fixed-point theory (Halder et al., 2022, Kakumani et al., 2022).
Representation in ODE/DAE Systems: For structured transit- or compartmental models, connections allow translation of PDE models into ODE systems (e.g., via reduction with gamma or uniform maturation time kernels), as well as state-dependent distributed delay DDEs (Cassidy et al., 2018).

6. Applications: Demography, Ecology, Epidemiology, and Beyond

The McKendrick–von Foerster equations underpin rigorous quantitative analysis in structured demography (age, size, or type):

Classical Demography: They yield precise connections to discrete-age Leslie matrix models, Euler–Lotka equilibrium relations, and can incorporate frequency-dependent selection via replicator–age–structured frameworks; this enables game-theoretic analyses reflecting sex ratio, strategy competition, and reproductive value dynamics (Argasinski et al., 2013).
Epidemic Modeling: Representing the age of infection as a structure variable provides a universal reduction for complex compartmental epidemic models, including SEIR-type and multi-type risk classes. The McKendrick–von Foerster representation captures arbitrary sojourn distributions and yields explicit renewal equations for inference and parameter estimation (Foutel-Rodier et al., 2020).
Cell Proliferation and Cycle Kinetics: Multi-compartmental Markov models of cell division converge to the McKendrick–von Foerster equation in the scaling limit. Explicit connections to spatially-extended traveling waves, reaction-diffusion models, and structured branching processes are established (Kynaston et al., 2021).
Growth-Fragmentation and Marine Ecosystems: In marine food-web models, the McKendrick–von Foerster equation (and its diffusive extension) encapsulates body-mass–structured size spectra. Rigorous spectral analysis reveals instability of power-law steady states in the absence of diffusion, and identifies mechanisms for restoration of stability (Datta et al., 2010).
Measure-theoretic Population Dynamics: Entropy-type methods and measure-valued solutions allow for analysis of singular or heterogeneous initial conditions and track convergence toward stable stationary distributions in a broad array of scalar and coupled models (Gwiazda et al., 2016).

7. Generalizations, Assumptions, and Limitations

Functional Requirements: Key assumptions on the rates (local boundedness, positivity, irreducibility/mixing conditions) determine well-posedness in $L^1$ or weighted spaces. Relaxing these conditions, e.g. to unbounded death rates or more singular fertility kernels, typically requires additional weighted norms or robust measure-space techniques (Mischler et al., 2013, Banasiak et al., 2022).
Boundary and Nonlocal Nonlinearity: Nonlinear and nonlocal birth boundary conditions (integral, history-dependent, environmental feedback) introduce analytical complications requiring refined fixed-point, Banach contraction, or threshold-based discretization analysis (Bartłomiejczyk et al., 2014, Halder et al., 2022).
Extensions: Spatial extension leads to age-structured reaction-diffusion equations. Generalization to multi-type, branching, or interacting populations leads to kinetic (BBGKY-type) or master equation hierarchies, with the mean-field McKendrick–von Foerster equation recovered only for population-independent rates (Chou et al., 2015, Greenman et al., 2015).
Spectral and Eigenstructure Limitations: In the presence of population- or environment-dependent rates, compactness and spectral gap properties may fail. Nonlinear models typically lack the explicit spectral representation and require alternative Lyapunov or entropy-based global analysis (Mischler et al., 2013, Gwiazda et al., 2016).

The McKendrick–von Foerster equations thus provide a rigorous, universal backbone for structured transport, renewal, and fragmentation processes in mathematical biology, with extensions spanning both continuum and measure-theoretic settings, and with deep connections to stochastic process hierarchies and modern numerical analysis (Mischler et al., 2013, Gwiazda et al., 2016, Carrillo et al., 2018, Banasiak et al., 2022, Greenman et al., 2015).