Age-Structured Chemostat Model

• Age-Structured Chemostat Model is a population-dynamical system that explicitly incorporates individual age profiles along with reactor dilution constraints.
• The model couples a first-order hyperbolic PDE with nonlocal boundary conditions and a nonlinear ODE for substrate dynamics to capture system behavior.
• Advanced output feedback laws, including sampled-data and continuous-time designs, ensure robust stabilization even under actuator limitations and model uncertainties.

An age-structured chemostat model is a class of population-dynamical control system that models the evolution and regulation of a microbial or cell population in a bioreactor, while explicitly accounting for the physiological age of individuals, the constraints of the reactor (e.g., dilution rate limits), and sometimes resource (substrate) dynamics. Modern formulations treat the problem as a coupled system of a first-order hyperbolic PDE with nonlocal boundary and a possible nonlinear ODE for substrate, equipped with robust and practically implementable output feedback laws for stabilization and control.

1. Mathematical Formulation of the Age-Structured Chemostat

The standard age-structured chemostat is modeled by the following McKendrick–von Foerster (Lotka–von Foerster) PDE, often coupled with substrate dynamics:

For $a\in[0,A]$ (age), $t\ge0$ (time), $x(t,a)\ge0$ (population density of age $a$ at time $t$), and $D(t)$ (dilution rate control input in $[D_{\min},D_{\max}]$):

$$\partial_t x(t,a) + \partial_a x(t,a) + \mu(a) x(t,a) = -D(t) x(t,a),\quad 0 < a < A, \, t>0$$

$x(t,0) = \int_0^A \beta(a) x(t,a) \, da$

$$x(0,a) = x_0(a) \ge 0,\,\, x_0(0) = \int_0^A \beta(a) x_0(a) da$$

Here, $\mu(a)$ is the (age-dependent) mortality rate, $\beta(a)$ the birth (fecundity) kernel, and $A>0$ the maximal age. The boundary at $a=0$ models renewal via reproduction. Constraints on $D(t)$ capture physical reactor limits.

When including substrate ($S(t)\ge0$), the associated ODE for substrate balance is

$$\dot S(t) = D(t)[S_{\rm in} - S(t)] - u(S(t)) \int_0^\infty q(a) n(a,t) da$$

with $n(a,t)$ the age-density, $u(S)$ specific growth (Monod/Haldane), $q(a)$ the per-cell substrate consumption kernel (Karafyllis et al., 13 Nov 2025).

2. Equilibrium Profiles and the Lotka–Sharpe Condition

Equilibria of the age-structured chemostat correspond to steady age profiles $x^*(a)$ and constant dilution $D^*$. The equilibrium age distribution must satisfy:

$$\frac{d}{da} x^*(a) + ( \mu(a) + D^* ) x^*(a) = 0,\quad x^*(0) = \int_0^A \beta(a) x^*(a) da$$

with general solution

$$x^*(a) = M \exp\Big( - D^* a - \int_0^a \mu(s) ds \Big)$$

for any $M > 0$.

The Lotka–Sharpe condition implicitly determines $D^*$: $$1 = \int_0^A \beta(a) \exp\Big( - D^* a - \int_0^a \mu(s) ds \Big) da$$ Existence of $D^* \in (D_{\min}, D_{\max})$ ensures a nontrivial equilibrium (Karafyllis et al., 2015).

For systems with substrate, similar equilibrium conditions require joint balance of age-profile and substrate concentration, often solvable via coupled nonlinear equations (Karafyllis et al., 13 Nov 2025, Karafyllis et al., 13 Feb 2025).

3. Integral Delay Representation and Well-Posedness

The McKendrick–von Foerster PDE with nonlocal boundary can be equivalently recast as a combination of a finite-dimensional ODE and an infinite-dimensional integral delay equation (IDE):

• Defining suitable functionals (e.g., $\Pi(x)$, “ergodic functional” II$(f)$, or moments $X(t), Y(t)$), and using the method of characteristics, the interior state evolution is “decoupled” into:

$\dot{n}(t) = D^* - D(t)$

$w(t) = \int_0^A k^*(a) w(t-a) da$

• This decoupling is fundamental both for analysis (Lyapunov functionals, contraction estimates) and feedback law design (Karafyllis et al., 2016, Haacker et al., 2023).

Global existence and uniqueness of solutions (“well-posedness”) for all admissible initial conditions and dilution rates $D(t)$ are established in the $L^1$ norm for age profiles and standard topology for $S(t)$, using Banach’s fixed-point theorem, a priori estimates, and contraction mappings (Karafyllis et al., 13 Nov 2025). The positivity of the state is guaranteed by the structure of the equations and the renewal boundary.

4. Output Feedback and Stabilization Laws

Modern feedback designs achieve global stabilization of the age-structured chemostat using only aggregate output measurements (not requiring full knowledge of the age profile). Two classes of bounded, observer-free feedback laws are central:

4.1. Sampled-Data Law

For arbitrary sampling period $T > 0$, using only $y(t_k)$ at sampling times $t_k = kT$:

$$D(t) = D_k := \operatorname{sat}_{[D_{\min}, D_{\max}]}\left( D^* + \frac{1}{T} \ln \frac{y(t_k)}{y^*} \right), \quad t_k \le t < t_{k+1}$$

where

$y(t) = \int_0^A p(a) x(t,a) da$

and

$y^* = \int_0^A p(a) x^*(a) da$

This law yields global exponential convergence in the sup-norm of the logarithmic relative error (Karafyllis et al., 2015, Karafyllis et al., 2016).

4.2. Continuous-Time Law

A dynamic observer-free feedback with internal states $(z_1, z_2)$ and only $y(t)$ sampled continuously:

$\dot{z}_1 = z_2 - D(t) - l_1 (z_1 - Y )$

$\dot{z}_2 = - l_2 (z_1 - Y )$

$$D(t) = \operatorname{sat}\{ z_2(t) + \gamma Y(t) \}$$

with $Y(t) = \ln(y(t)/y^*)$. The gain parameters $l_1, l_2, \gamma$ can be tuned. This design ensures $D(t) \in [D_{\min}, D_{\max}]$ and global asymptotic stability of the desired profile (Karafyllis et al., 2016).

4.3. Robustness, Sparse Sampling, and Model Uncertainty

Both sampled-data and continuous-time designs are robust to sparse sampling (possibly arbitrarily large $T$), and only require approximate knowledge of $D^*$. Small errors in $D^*$ yield bounded bias in steady-state. Absence of full age-profile measurements does not degrade stability (Karafyllis et al., 2015, Karafyllis et al., 2016). Input constraints $[D_{\min}, D_{\max}]$ are always enforced.

5. Extensions: Substrate Dynamics and Moment-Closure

When the growth rate depends on a limiting resource (substrate), the system couples the age-PDE with a nonlinear ODE for $S(t)$. The general model is:

$$\frac{\partial n}{\partial t}(a,t) + \frac{\partial n}{\partial a}(a,t) = -[ \mu(a) + D(t) ] n(a,t )$$

$n(0,t) = u(S(t)) \int_0^\infty k(a) n(a,t) da$

$$\dot S(t) = D(t) ( S_{\rm in} - S(t) ) - u(S(t)) \int_0^\infty q(a) n(a,t) da$$

Well-posedness for these coupled PDE-ODE systems has been established globally under standard regularity and positivity hypotheses (Karafyllis et al., 13 Nov 2025). In special cases, a moment-closure leads to finite-dimensional ODE systems for quantities like total substrate consumption and population activity, facilitating analysis and feedback design.

Explicit feedback mechanisms for the age-structured chemostat with substrate have been constructed to guarantee global stabilization, subject to positivity and input constraints even in the presence of nonzero natural mortality rates (Karafyllis et al., 13 Feb 2025). The controller fortifies classical feedback with terms to prevent biomass extinction when mortality is significant.

6. Models with Actuator Dynamics and Saturated Control

Practical implementations often require modeling actuator limitations (e.g., dilutor inertia), introducing additional state $D(t)$ governed by integrator dynamics $\dot D(t) = u(t)$. The age-structured PDE then couples to actuator ODEs:

$\dot D(t) = u(t)$

Stabilizing controllers must now take actuator constraints into account.

Advanced Lyapunov-based backstepping and mode-decoupling techniques allow for robust stabilization even under actuator dynamics and saturation ($D(t) \in [D_{\min}, D_{\max}]$). Both full-state and (practically relevant) output-based feedbacks are available. Positive safety filters and control barrier functions ensure $D(t)$ remains positive, and saturated continuous controllers achieve global asymptotic (KL class) convergence (Haacker et al., 2023).

Lyapunov functionals for these analyses typically involve combinations of log-coordinates, characteristic-based integral delay states, and weighted sup-norms.

7. Applications, Special Cases, and Dynamics Beyond ODE Models

The age-structured chemostat captures population phenomena inaccessible to lumped ODE models:

• Oscillatory (limit cycle) dynamics can arise even in single-species age-structured systems, as noted by Tõth and Kot, which classical chemostats cannot reproduce (Karafyllis et al., 13 Nov 2025).
• In the case where all aging kernels are Dirac masses or piecewise-constant, the age-PDE reduces to a system of delay-differential equations.
• When $u(S)$ is Monod and the system is linearized near steady state, stability conditions recover and extend those of classical chemostats with adjustment by age-structure (Karafyllis et al., 13 Nov 2025).

These models and control laws form a rigorous basis for feedback-stabilized operation of bioreactors and have opened the way for further robust, adaptive, and optimal control research in structured population systems.