Fractional-Order Chemostat Model (FOCM)

• FOCM is a fractional-differential model that incorporates memory effects and periodic operation to describe substrate and biomass dynamics in water treatment.
• It employs a Caputo fractional derivative with a sliding memory window and Contois kinetics, achieving up to a 40% reduction in average substrate concentration.
• The model guarantees existence, uniqueness, positivity, and bang-bang optimal control, with numerical methods mitigating challenges like Gibbs oscillations.

The Fractional-Order Chemostat Model (FOCM) is a chemostat formulation in which substrate and biomass dynamics are governed by fractional differential operators rather than ordinary derivatives, with the explicit purpose of representing memory effects, periodic operation, and time-dependent microbial adaptation in biological water treatment. In the recent control-oriented formulation, the model uses the Caputo fractional derivative with sliding memory (CFDS), periodic boundary conditions (PBCs), and Contois growth kinetics to describe pollutant removal under periodic dilution-rate modulation. Within that framework, the primary optimization target is the minimization of the average substrate concentration over one cycle, and the reported numerical results include reductions of average substrate concentration by up to $40\%$ relative to steady-state operation (Elgindy et al., 1 Aug 2025). A closely related mathematical framework establishes existence, uniqueness, positivity, boundedness, and stability of periodic solutions for the sliding-memory fractional chemostat itself, thereby supplying the well-posedness theory on which the control problem rests (Elgindy, 4 Jun 2025).

1. Fractional formulation and periodic-memory structure

The state variables are the substrate concentration $s(t)$ or $S(t)$, the biomass concentration $x(t)$ or $X(t)$, and, in the control setting, the dilution rate $D(t)$, with bounds $D_{\min}\le D(t)\le D_{\max}$. The inflow substrate concentration is $s_{in}$ or $S_{in}$, the yield coefficient is $Y>0$, the fractional order satisfies $s(t)$0, and the sliding memory length satisfies $s(t)$1. The control-oriented model also introduces a dynamic scaling parameter $s(t)$2, appearing as $s(t)$3, which modulates the amplitude of the fractional dynamics. The specific growth law is the Contois model,

$s(t)$4

with $s(t)$5 and $s(t)$6 (Elgindy et al., 1 Aug 2025).

The CFDS replaces the fixed-origin history integral of the classical Caputo derivative by a finite sliding window. The left-sided operator is

$s(t)$7

and the right-sided operator used in the adjoint system is

$s(t)$8

With PBCs, the full fractional-order chemostat system is

$s(t)$9

$S(t)$0

together with

$S(t)$1

A key periodicity-preserving identity is

$S(t)$2

This finite-window construction is central: unlike the standard Caputo derivative integrating from a fixed initial time, CFDS is designed to preserve periodicity of $S(t)$3-periodic absolutely continuous functions, and the parameters $S(t)$4, $S(t)$5, and $S(t)$6 jointly shape memory effects, responsiveness, and the magnitude of the dynamics (Elgindy et al., 1 Aug 2025, Elgindy, 4 Jun 2025).

2. Relation to the classical chemostat and reduction to a one-dimensional FDE

The baseline integer-order chemostat model is written as

$S(t)$7

The fractional formulation modifies these balances by replacing $S(t)$8 with a sliding-memory fractional operator and, in the control formulation, multiplying the right-hand side by $S(t)$9. This introduces non-local dependence into the growth and washout terms and changes the transient regime from memoryless ODE dynamics to hereditary FDE dynamics (Elgindy et al., 1 Aug 2025).

A defining structural result is the reduction of the two-dimensional system to a one-dimensional fractional differential equation. The transformation

$x(t)$0

yields

$x(t)$1

in the control setting, or $x(t)$2 in the foundational formulation. Under the periodicity assumptions and the energy argument given in the analysis, there are no non-trivial periodic solutions for $x(t)$3, hence

$x(t)$4

The reduced substrate equation becomes

$x(t)$5

with

$x(t)$6

In the corresponding non-scaled framework the same reduction gives

$x(t)$7

This reduction is computationally decisive because it preserves the essential substrate dynamics needed for optimization while eliminating the second state through the exact identity $x(t)$8 (Elgindy et al., 1 Aug 2025, Elgindy, 4 Jun 2025).

For constant dilution $x(t)$9, the reduced equation has a unique interior equilibrium satisfying $X(t)$0, namely

$X(t)$1

The alternative equilibrium is the washout state $X(t)$2 (Elgindy, 4 Jun 2025).

3. Existence, uniqueness, positivity, and stability

The periodic theory is developed in the space $X(t)$3 of absolutely continuous $X(t)$4-periodic functions. A standard feasible state set is

$X(t)$5

and admissible controls are $X(t)$6-periodic $X(t)$7 functions satisfying the average constraint and pointwise bounds. In the foundational analysis, the right-hand side satisfies Carathéodory regularity, and the reduced problem is rewritten as a Volterra integral equation over the sliding window. The induced fixed-point operator is continuous, maps the compact convex set into itself, and has relatively compact image, so Schauder’s fixed-point theorem yields existence of at least one $X(t)$8-periodic Carathéodory solution (Elgindy, 4 Jun 2025).

The control-oriented analysis proves that, under $X(t)$9, $D(t)$0, and $D(t)$1 for all $D(t)$2, the reduced fractional optimal periodic control problem has at least one optimal periodic solution $D(t)$3. The argument uses compactness and convexity of the feasible state set, weak-$D(t)$4 compactness of the admissible control set via Banach–Alaoglu, continuity of the control-to-state map through the Volterra formulation, and continuity of the objective functional $D(t)$5 (Elgindy et al., 1 Aug 2025).

Positivity and boundedness are built into the reduced dynamics. At $D(t)$6, the drift satisfies $D(t)$7, and at $D(t)$8, $D(t)$9, which enforces $D_{\min}\le D(t)\le D_{\max}$0. Because $D_{\min}\le D(t)\le D_{\max}$1, biomass remains nonnegative, and if $D_{\min}\le D(t)\le D_{\max}$2 then $D_{\min}\le D(t)\le D_{\max}$3 for all $D_{\min}\le D(t)\le D_{\max}$4. The biologically feasible invariant set is

$D_{\min}\le D(t)\le D_{\max}$5

For stability, the reduced one-dimensional dynamics $D_{\min}\le D(t)\le D_{\max}$6 satisfy

$D_{\min}\le D(t)\le D_{\max}$7

so linearization gives algebraic decay governed by the Mittag–Leffler function. In the full linear fractional system, asymptotic stability requires eigenvalues $D_{\min}\le D(t)\le D_{\max}$8 of the Jacobian to satisfy

$D_{\min}\le D(t)\le D_{\max}$9

At washout, stability holds for $s_{in}$0 and fails for $s_{in}$1 (Elgindy, 4 Jun 2025).

Uniqueness is conditional rather than universal. In the foundational paper, uniqueness of the $s_{in}$2-periodic Carathéodory solution is established under any of three stated conditions, including $s_{in}$3, $s_{in}$4, and $s_{in}$5 for all $s_{in}$6, or conditions involving

$s_{in}$7

and upper bounds on $s_{in}$8 or $s_{in}$9 by $S_{in}$0. In the control problem, uniqueness of the optimal periodic control is proved under $S_{in}$1 and additional conditions such as

$S_{in}$2

together with either $S_{in}$3 for all $S_{in}$4 or $S_{in}$5 (Elgindy, 4 Jun 2025, Elgindy et al., 1 Aug 2025).

4. Optimal periodic control and bang-bang structure

The optimal periodic control problem minimizes the cycle-averaged substrate concentration,

$S_{in}$6

subject to

$S_{in}$7

the state and control bounds

$S_{in}$8

and the periodicity conditions on $S_{in}$9, $Y>0$0, and $Y>0$1. The model allows non-constant optimal controls, and the paper states that if $Y>0$2 and $Y>0$3, then non-constant optimal periodic solutions may exist with improved average substrate concentration compared to steady-state operation. The perturbation analysis distinguishes three regimes: for $Y>0$4, concavity of $Y>0$5 implies $Y>0$6, enabling $Y>0$7; for $Y>0$8, improvements are possible but not guaranteed; for $Y>0$9, averages match steady-state (Elgindy et al., 1 Aug 2025).

The fractional Pontryagin maximum principle yields the Hamiltonian

$s(t)$00

the state equation

$s(t)$01

and the right-sided adjoint equation

$s(t)$02

with periodic transversality $s(t)$03. The switching function is

$s(t)$04

Since $s(t)$05 and $s(t)$06, the optimal control satisfies the bang-bang law

$s(t)$07

Singular arcs are ruled out because $s(t)$08 would imply $s(t)$09, contradicting the adjoint equation $s(t)$10. Accordingly, the optimal control is bang-bang, and the number of switches per period is even due to periodicity (Elgindy et al., 1 Aug 2025).

5. Numerical realization and reported performance

The reported implementation uses a Fourier–Gegenbauer pseudospectral (FG-PS) method on $s(t)$11 equispaced collocation points. The CFDS term is approximated through precomputed integration matrices tailored to periodic problems, and the discretized reduced control problem is solved as a constrained NLP by MATLAB’s fmincon with the sqp algorithm. Because the pseudospectral approximation of a bang-bang profile exhibits Gibbs oscillations, the workflow includes an edge-detection correction: Fourier coefficients of the predicted $s(t)$12 are computed, discontinuities are detected on the spectral profile, the bang-bang control is reconstructed from the detected switches, and the state $s(t)$13 is corrected with fsolve using the reconstructed $s(t)$14 in a predictor-corrector approach (Elgindy et al., 1 Aug 2025).

Typical parameters used in the examples are $s(t)$15–$s(t)$16 collocation points, $s(t)$17–$s(t)$18 interpolation points, $s(t)$19, $s(t)$20, $s(t)$21, $s(t)$22, $s(t)$23, $s(t)$24, $s(t)$25, $s(t)$26, $s(t)$27, $s(t)$28, and $s(t)$29. In the base case, the average substrate concentration is reduced from the steady-state value $s(t)$30 to $s(t)$31, corresponding to an approximately $s(t)$32 improvement. With $s(t)$33, the paper reports $s(t)$34 versus steady-state $s(t)$35, a $s(t)$36 reduction, described as nearly $s(t)$37 (Elgindy et al., 1 Aug 2025).

The sensitivity analysis attributes distinct operational roles to $s(t)$38, $s(t)$39, $s(t)$40, and $s(t)$41. Lower $s(t)$42 produces stronger memory and more switches per period; higher $s(t)$43 yields fewer switches, while preserving the bang-bang structure. The function $s(t)$44 is non-monotonic for small $s(t)$45, with a peak around $s(t)$46, and decreases monotonically for $s(t)$47. As $s(t)$48 increases, $s(t)$49 first rises slightly from $s(t)$50 to $s(t)$51, then decreases up to approximately $s(t)$52, with diminishing returns beyond that range. Increasing $s(t)$53 monotonically decreases $s(t)$54, and increasing the period $s(t)$55 over the tested range $s(t)$56–$s(t)$57 also decreases $s(t)$58. Because the optimal control is discontinuous, convergence of the state approximation is algebraic rather than spectral, although the switching times are reported as accurate across resolutions to machine precision (Elgindy et al., 1 Aug 2025).

6. Interpretation, limitations, and adjacent research directions

The FOCM interprets $s(t)$59 and $s(t)$60 as descriptors of microbial adaptation latency and non-local response, while $s(t)$61 tunes dynamic responsiveness. The periodic boundary conditions and the periodicity-preserving structure of CFDS align the mathematics with cyclic operational strategies such as scheduled dilution or diurnal loading. The foundational analysis states the principal assumptions explicitly: perfect mixing, constant $s(t)$62, stationary parameters $s(t)$63, Contois kinetics, periodic operation, and a finite sliding memory window representing localized historical dependence (Elgindy et al., 1 Aug 2025, Elgindy, 4 Jun 2025).

The principal limitations are equally explicit. The model is single-species and single-substrate; parameters are assumed constant and identifiable; fractional operators and non-smooth controls introduce computational overhead; Gibbs phenomena require numerical mitigation; and uniqueness is only guaranteed under specific conditions, so multiple local optima may arise in the general case. The stated practical guidelines are correspondingly operational: choose $s(t)$64 close to $s(t)$65 for faster-responding communities and fewer control switches; use a moderately large $s(t)$66 to capture useful memory without excessive computational burden; and increase $s(t)$67 to enhance responsiveness and reduce $s(t)$68, while balancing actuator limits and system stability. Proposed extensions include multi-species and multi-substrate models, inhibition kinetics such as Haldane kinetics, stochastic forcing through uncertain influent $s(t)$69 or random shocks, adaptive or robust fractional control, and distributed systems based on fractional PDEs (Elgindy et al., 1 Aug 2025, Elgindy, 4 Jun 2025).

A recurrent misconception in the broader fractional-compartment literature is that fractional modeling is exhausted by replacing an integer-order derivative with a fractional one. A related stochastic study argues instead for a physically grounded construction in which fractional behavior arises from semi-Markov dynamics and Mittag–Leffler waiting times, yielding Riemann–Liouville operators at the mean-field level and exact SSA realizations for small populations. That work also emphasizes that deterministic fractional equations can diverge substantially from stochastic dynamics when the population is small. This suggests that future FOCM research may need to distinguish more sharply between phenomenological sliding-memory formulations for periodic bioprocess control and stochastic semi-Markov derivations for chemostat systems operating far from the continuum limit (Angstmann et al., 2023).