The reviewed record of science sign in
Pith

arxiv: 2607.01347 · v1 · pith:2TWCXUND · submitted 2026-07-01 · math.OC

Bilinear control of age--space structured populations

Jiguang Yu , Louis Shuo Wang This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-03 19:27 UTCgrok-4.3pith:2TWCXUNDrecord.jsonopen to challenge →

classification math.OC
keywords bilinear optimal controlage-space structured populationsrenewal boundary conditionsendogenous surveillance feedbackcharacteristic mild formulationfeedback-corrected adjointswitching function decompositionpopulation dynamics
0
0 comments X

The pith

A characteristic mild formulation establishes well-posedness and first-order optimality conditions for bilinear control of age-space structured populations with endogenous feedback.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies bilinear optimal control of population equations structured by age and space, where the control multiplies a mixed transport-diffusion term and a scalar state observable feeds back into both the interior equation and the renewal boundary condition. It replaces standard weak-solution arguments with a characteristic mild formulation to prove that the resulting nonlinear closed-loop control-to-state map remains well-posed and Fréchet differentiable. From this map the authors obtain a feedback-corrected adjoint system whose perturbation is low-rank, derive an explicit resolvent representation when the kernel is Volterra, and prove first-order optimality conditions that split the switching function into reduced and feedback-induced parts.

Core claim

Using a characteristic mild formulation rather than a standard Lions-Magenes argument, the authors establish closed-loop well-posedness and Fréchet differentiability of the control-to-state map, derive the feedback-corrected adjoint equations, and prove first-order optimality conditions with an explicit decomposition of the switching function into reduced and feedback-induced components for bilinear control of nonlocal age-space structured population equations with renewal boundary conditions and endogenous surveillance feedback.

What carries the argument

The characteristic mild formulation of the nonlocal transport-diffusion equation with renewal boundary conditions, which accommodates the low-rank feedback perturbation and yields the closed-loop well-posedness and differentiability results.

Load-bearing premise

The endogenous surveillance feedback produces a well-defined scalar observable that enters both the interior dynamics and the renewal law in a manner compatible with the characteristic mild formulation.

What would settle it

An explicit example of an observable generated by the state for which the closed-loop system fails to be well-posed or for which the switching-function decomposition does not hold under the stated control constraints.

Figures

Figures reproduced from arXiv: 2607.01347 by Jiguang Yu, Louis Shuo Wang.

Figure 1
Figure 1. Figure 1: Characteristic transport vs. spatial diffusion. The figure contrasts the hyperbolic age-time transport along characteristics (𝑡 − 𝑎 = const), which exhibits no smoothing, with the parabolic spatial diffusion driven by the evolution family 𝑈(𝑎, 𝜎) in the 𝑥-direction. This structural difference clarifies the nature of the energy space  and explains the bypass of the standard Lions–Magenes continuity argumen… view at source ↗
Figure 2
Figure 2. Figure 2: The closed-loop feedback architecture. This operator block diagram maps the nonlinear control-to-state map and highlights the endogenous feedback closure. Distinct pathways illustrate the state variables (𝑦), the endogenous observable (𝐸𝑦 (𝑡)), and the bilinear control (𝑢). Tracking these exact dependencies of the fixed-point mapping 𝑦 = T𝑢 (𝐸, 𝑦) justifies the weighted Volterra contraction estimate via th… view at source ↗
Figure 3
Figure 3. Figure 3: The Backward Volterra adjoint and low-rank perturbation. The diagram displays the backward-triangular integration domain △𝑇 = {0 ≤ 𝑡 ≤ 𝑠 ≤ 𝑇 }, contrasting the forward causal reduced operator  with the anti-causal adjoint  ∗ . A localized rank-one excitation depicts the structural correction 𝓁̄𝑦, ̄𝑢(𝑝)(𝑠)𝜒, visually demonstrating why the strict triangular support eliminates diagonal singularities and gua… view at source ↗
Figure 4
Figure 4. Figure 4: Threshold preservation in the switching function. This graphical representation of Theorem 5.10 illustrates how the feedback perturbation alters the optimal intervention decision. The uncertainty band is defined by ±|𝑆f b|. The shaded decision-change set f b is strictly confined inside the region where |𝑆red| ≤ |𝑆f b|. Outside this narrow margin, the sign of the switching function (and thus the optimal ba… view at source ↗
read the original abstract

We study constrained bilinear optimal control for nonlocal age--space structured population equations with renewal boundary conditions and endogenous surveillance feedback. The control acts as a coefficient in a mixed transport--diffusion equation, while a scalar observable generated by the state enters both the interior dynamics and the renewal law. This produces a nonlinear closed-loop control-to-state map and a feedback-dependent adjoint system. Using a characteristic mild formulation rather than a standard Lions--Magenes argument, we establish closed-loop well-posedness and Frechet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation $\ell_{\bar y,\bar u}(p)(t)\chi(a,x)$; in the Volterra-kernel regime, the associated transfer operator is quasinilpotent, yielding an explicit resolvent representation of the adjoint. Finally, we prove first-order optimality conditions and decompose the switching function into reduced and feedback-induced components.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies constrained bilinear optimal control for nonlocal age-space structured population equations with renewal boundary conditions and endogenous surveillance feedback. The control enters as a coefficient in a mixed transport-diffusion equation, while a scalar observable generated by the state enters both the interior dynamics and the renewal law. Using a characteristic mild formulation, the authors claim to establish closed-loop well-posedness and Fréchet differentiability of the control-to-state map, derive the feedback-corrected adjoint as a low-rank perturbation, obtain an explicit resolvent representation when the transfer operator is quasinilpotent, and prove first-order optimality conditions with an explicit decomposition of the switching function into reduced and feedback-induced components.

Significance. If the central claims hold, the work advances the analysis of feedback-dependent bilinear control problems in structured population models by providing an explicit resolvent for the adjoint and a decomposition of the switching function. The methodological choice of the characteristic mild formulation (avoiding Lions-Magenes arguments) is a technical strength that could extend to other nonlocal renewal problems in mathematical biology and control theory.

major comments (2)
  1. [Abstract and well-posedness theorem (characteristic mild formulation)] The closed-loop well-posedness and Fréchet differentiability claims rest on the scalar observable ℓ_{ar y,ar u}(p)(t) producing a low-rank perturbation ℓ_{ar y,ar u}(p)(t)χ(a,x) whose Volterra kernel satisfies the quasinilpotence needed for the resolvent representation. No explicit regularity or boundedness conditions on this observable (e.g., L^∞ bounds, trace regularity, or compatibility with the characteristic curves) are stated; this assumption is load-bearing for the perturbation to remain controllable and for the mild formulation to close.
  2. [Optimality conditions and adjoint derivation] The decomposition of the switching function into reduced and feedback-induced components in the first-order optimality conditions depends on the explicit resolvent of the feedback-corrected adjoint. Without the missing function-space assumptions on the observable, the quasinilpotence of the transfer operator and the validity of this decomposition cannot be verified from the given setup.
minor comments (2)
  1. [Abstract] The abstract is dense; a brief sentence clarifying the precise function spaces (e.g., the underlying Banach space for the age-space density) would improve readability.
  2. [Notation and setup] Notation for the observable and the low-rank perturbation χ(a,x) would benefit from an early dedicated definition or table of symbols.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicit statement of regularity assumptions on the observable, which we will clarify in revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and well-posedness theorem (characteristic mild formulation)] The closed-loop well-posedness and Fréchet differentiability claims rest on the scalar observable ℓ_{ar y,ar u}(p)(t) producing a low-rank perturbation ℓ_{ar y,ar u}(p)(t)χ(a,x) whose Volterra kernel satisfies the quasinilpotence needed for the resolvent representation. No explicit regularity or boundedness conditions on this observable (e.g., L^∞ bounds, trace regularity, or compatibility with the characteristic curves) are stated; this assumption is load-bearing for the perturbation to remain controllable and for the mild formulation to close.

    Authors: We agree that the manuscript does not list explicit function-space assumptions on the observable in the well-posedness theorem or abstract. The observable is defined via a linear functional of the state in the model setup, and the characteristic mild formulation in Section 3 derives a priori L^∞ bounds from the transport-diffusion structure and renewal conditions. To make the load-bearing assumption transparent and allow verification of quasinilpotence, we will add an explicit hypothesis (A3) on the observable's regularity and boundedness in the revised version. revision: yes

  2. Referee: [Optimality conditions and adjoint derivation] The decomposition of the switching function into reduced and feedback-induced components in the first-order optimality conditions depends on the explicit resolvent of the feedback-corrected adjoint. Without the missing function-space assumptions on the observable, the quasinilpotence of the transfer operator and the validity of this decomposition cannot be verified from the given setup.

    Authors: The decomposition is derived under the Volterra-kernel regime where the transfer operator is quasinilpotent, as stated in the paper. We acknowledge that the function-space assumptions needed to guarantee this quasinilpotence from the observable are not stated explicitly enough for independent verification. We will revise the hypotheses of the optimality theorem to include these assumptions, ensuring the resolvent representation and switching-function decomposition can be checked directly from the setup. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external functional-analytic tools

full rationale

The paper establishes closed-loop well-posedness, Fréchet differentiability, and first-order optimality conditions for the bilinear control problem via a characteristic mild formulation of the nonlocal age-space equations. These steps invoke standard results on Volterra kernels, quasinilpotence of transfer operators, and low-rank perturbations without reducing any claimed prediction or uniqueness statement to quantities defined by the authors' own fitted parameters or prior self-citations. The scalar observable and feedback terms are treated as given inputs compatible with the mild formulation; no equation is shown to be equivalent to its own inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard existence theory for mild solutions of transport-diffusion equations and differentiability of control-to-state maps in infinite-dimensional spaces, but supplies no explicit list of free parameters or invented entities.

axioms (2)
  • domain assumption The characteristic mild formulation yields a well-posed closed-loop system for the given class of nonlocal renewal boundary conditions.
    Invoked to replace Lions-Magenes arguments and to obtain Fréchet differentiability.
  • domain assumption The feedback derivative appears as a low-rank perturbation whose transfer operator is quasinilpotent in the Volterra-kernel regime.
    Used to obtain an explicit resolvent representation of the adjoint.

pith-pipeline@v0.9.1-grok · 5686 in / 1247 out tokens · 16072 ms · 2026-07-03T19:27:20.396348+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    A unified approach to abstract linear nonautonomous parabolic equations

    Acquistapace, P., Terreni, B., 1987. A unified approach to abstract linear nonautonomous parabolic equations. Rendiconti del seminario matematico della Università di Padova 78, 47–107

    work page 1987

  2. [2]

    Two-compartment age-structured model of solitarious and gregarious locust population dynamics

    Akimenko, V.V., Piou, C., 2018. Two-compartment age-structured model of solitarious and gregarious locust population dynamics. Mathematical Methods in the Applied Sciences 41, 8636–8672

    work page 2018

  3. [3]

    Analysis and control of age-dependent population dynamics

    Anita, S., 2013. Analysis and control of age-dependent population dynamics. volume 11. Springer Science & Business Media

    work page 2013

  4. [4]

    Mathematical methods in optimization of differential systems

    Barbu, V., 2012. Mathematical methods in optimization of differential systems. volume 310. Springer Science & Business Media

    work page 2012

  5. [5]

    On the formulation of size-structured consumer resource models (with special attention for the principle of linearized stability)

    Barril, C., Calsina, À., Diekmann, O., Farkas, J.Z., 2022. On the formulation of size-structured consumer resource models (with special attention for the principle of linearized stability). Mathematical Models and Methods in Applied Sciences 32, 1141–1191

    work page 2022

  6. [6]

    On hierarchical competition through reduction of individual growth

    Barril, C., Calsina, À., Diekmann, O., Farkas, J.Z., 2024. On hierarchical competition through reduction of individual growth. Journal of mathematical biology 88, 66

    work page 2024

  7. [7]

    Existence and uniqueness of solutions for single-population mckendrick-von foerster models with renewal

    Bartłomiejczyk, A., Leszczyński, H., Zwierkowski, P., 2015. Existence and uniqueness of solutions for single-population mckendrick-von foerster models with renewal. The Rocky Mountain Journal of Mathematics 45, 401–426

    work page 2015

  8. [8]

    Functional analysis, Sobolev spaces and partial differential equations

    Brezis, H., Brézis, H., 2011. Functional analysis, Sobolev spaces and partial differential equations. volume 2. Springer

    work page 2011

  9. [9]

    Pontryagin’s principle for control problems in age-dependent population dynamics

    Brokate, M., 1985. Pontryagin’s principle for control problems in age-dependent population dynamics. Journal of Mathematical Biology 23, 75–101

    work page 1985

  10. [10]

    A model of physiologically structured population dynamics with a nonlinear individual growth rate

    Calsina, À., Saldaña, J., 1995. A model of physiologically structured population dynamics with a nonlinear individual growth rate. Journal of Mathematical Biology 33, 335–364

    work page 1995

  11. [11]

    An introduction to structured population dynamics

    Cushing, J.M., 1998. An introduction to structured population dynamics. SIAM

    work page 1998

  12. [12]

    On the formulation and analysis of general deterministic structured population models ii

    Diekmann, O., Gyllenberg, M., Huang, H., Kirkilionis, M., Metz, J., Thieme, H.R., 2001. On the formulation and analysis of general deterministic structured population models ii. nonlinear theory: Ii. nonlinear theory. Journal of mathematical biology 43, 157–189

    work page 2001

  13. [13]

    Steady-state analysis of structured population models

    Diekmann, O., Gyllenberg, M., Metz, J., 2003. Steady-state analysis of structured population models. Theoretical population biology 63, 309–338. J. Yu and L. S. Wang:Preprint submitted to ElsevierPage 23 of 25 Bilinear control of age–space structured populations

    work page 2003

  14. [14]

    On the formulation and analysis of general deterministic structured population models i

    Diekmann, O., Gyllenberg, M., Metz, J., Thieme, H.R., 1998. On the formulation and analysis of general deterministic structured population models i. linear theory: I. linear theory. Journal of Mathematical Biology 36, 349–388

    work page 1998

  15. [15]

    Finite dimensional state representation of physiologically structured populations

    Diekmann, O., Gyllenberg, M., Metz, J.A., 2020a. Finite dimensional state representation of physiologically structured populations. Journal of Mathematical Biology 80, 205–273

    work page

  16. [16]

    On models of physiologically structured populations and their reduction to ordinary differential equations

    Diekmann, O., Gyllenberg, M., Metz, J.A., 2020b. On models of physiologically structured populations and their reduction to ordinary differential equations. Journal of mathematical biology 80, 189–204

    work page

  17. [17]

    Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

    Diekmann, O., Gyllenberg, M., Metz, J.A., Nakaoka, S., de Roos, A.M., 2010. Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. Journal of mathematical biology 61, 277–318

    work page 2010

  18. [18]

    Anintegratedsemigroupapproachforagestructuredequationswithdiffusionandnon-homogeneous boundary conditions

    Ducrot,A.,Magal,P.,Thorel,A.,2021. Anintegratedsemigroupapproachforagestructuredequationswithdiffusionandnon-homogeneous boundary conditions. Nonlinear Differential Equations and Applications NoDEA 28, 49

    work page 2021

  19. [19]

    Size-specificsensitivity:applyinganewstructuredpopulationmodel

    Easterling,M.R.,Ellner,S.P.,Dixon,P.M.,2000. Size-specificsensitivity:applyinganewstructuredpopulationmodel. Ecology81,694–708

    work page 2000

  20. [20]

    Selectivity of invasive species suppression efforts influences control efficacy

    Fadlovich, R., Landom, K., Walsworth, T.E., 2025. Selectivity of invasive species suppression efforts influences control efficacy. Journal of Applied Ecology 62, 280–289

    work page 2025

  21. [21]

    Optimality conditions for age-structured control systems

    Feichtinger, G., Tragler, G., Veliov, V.M., 2003. Optimality conditions for age-structured control systems. Journal of Mathematical Analysis and Applications 288, 47–68

    work page 2003

  22. [22]

    Mathematicalanalysisofanon-convexoptimalcontrolproblemforage-structuredmosquitopopulations

    Filho,C.A.d.S.,Boldrini,J.L.,2025. Mathematicalanalysisofanon-convexoptimalcontrolproblemforage-structuredmosquitopopulations. Mathematical Methods in the Applied Sciences 48, 1381–1410

    work page 2025

  23. [23]

    Gao,Y.,Li,L.,Yu,J.,2022. Rollingpredictionmodelofclosingpricebasedoneemddatanoisereductionandhgs-delm,in:2022International Conference on Data Analytics, Computing and Artificial Intelligence (ICDACAI), IEEE. pp. 255–260

    work page 2022

  24. [24]

    Theory and applications of Volterra operators in Hilbert space

    Gohberg, I., Krein, M.G., 1970. Theory and applications of Volterra operators in Hilbert space. volume 24. American Mathematical Soc

    work page 1970

  25. [25]

    Non-linear age-dependent population dynamics

    Gurtin, M.E., MacCamy, R.C., 1974. Non-linear age-dependent population dynamics. Archive for Rational Mechanics and Analysis 54, 281–300

    work page 1974

  26. [26]

    Optimization with PDE constraints

    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S., 2008. Optimization with PDE constraints. Springer Science & Business Media

    work page 2008

  27. [27]

    Mathematical theory of age-structured population dynamics

    Iannelli, M., 1995. Mathematical theory of age-structured population dynamics. Giardini editori e stampatori in Pisa

    work page 1995

  28. [28]

    The basic approach to age-structured population dynamics

    Iannelli, M., Milner, F., et al., 2017. The basic approach to age-structured population dynamics. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Dordrecht

    work page 2017

  29. [29]

    Optimal harvesting for nonlinear size-structured populations with nonlocal environmental feedback

    Jie, C., Chen, X., Longfei, G., Jiayao, C., Nuo, C., Wang, L.S., Ye, L., Jiguang, Y., 2026. Optimal harvesting for nonlinear size-structured populations with nonlocal environmental feedback. Mathematics 14, 2025

    work page 2026

  30. [30]

    Global dynamics and asymptotic spreading of a diffusive age-structured model in spatially periodic media: H

    Kang, H., Liu, S., 2025. Global dynamics and asymptotic spreading of a diffusive age-structured model in spatially periodic media: H. kang, s. liu. Mathematische Annalen 393, 2631–2682

    work page 2025

  31. [31]

    The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application

    Kang, H., Peng, R., Zhou, M., 2026. The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application. Nonlinearity 39, 025008

    work page 2026

  32. [32]

    Age-structuredpopulationdynamicswithnonlocaldiffusion

    Kang,H.,Ruan,S.,Yu,X.,2022. Age-structuredpopulationdynamicswithnonlocaldiffusion. Journalofdynamicsanddifferentialequations 34, 789–823

    work page 2022

  33. [33]

    Measure-valued optimal control for size-structured population models with diffusion

    Kato, N., 2024. Measure-valued optimal control for size-structured population models with diffusion. Journal of Optimization Theory and Applications 201, 54–74

    work page 2024

  34. [34]

    Stabilityandoptimalcontrolofage-structuredcell-freeandcell-to-celltransmissionmodelofhiv

    Kumar,M.,Abbas,S.,2023. Stabilityandoptimalcontrolofage-structuredcell-freeandcell-to-celltransmissionmodelofhiv. Mathematical Methods in the Applied Sciences 46, 18336–18353

    work page 2023

  35. [35]

    A nonlinear problem in age-dependent population diffusion

    Langlais, M., 1985. A nonlinear problem in age-dependent population diffusion. SIAM Journal on Mathematical Analysis 16, 510–529

    work page 1985

  36. [36]

    Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

    Langlais, M., 1988. Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion. Journal of mathematical biology 26, 319–346

    work page 1988

  37. [37]

    Global well-posedness and stability of nonlocal damage-structured lineage model with feedback and dedifferentiation

    Liang, Y., Wang, L.S., Yu, J., Liu, Z., 2025. Global well-posedness and stability of nonlocal damage-structured lineage model with feedback and dedifferentiation. Mathematics 13, 3583

    work page 2025

  38. [38]

    Optimal control of systems governed by partial differential equations

    Lions, J.L., 1971. Optimal control of systems governed by partial differential equations. volume 170. Springer

    work page 1971

  39. [39]

    Non-homogeneousboundaryvalueproblemsandapplications:Vol.1.volume1

    Lions,J.L.,Magenes,E.,2012. Non-homogeneousboundaryvalueproblemsandapplications:Vol.1.volume1. SpringerScience&Business Media

    work page 2012

  40. [40]

    Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study

    Liu, Z., Wang, L.S., Yu, J., Zhang, J., Martel, E., Li, S., 2025. Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study. Bioengineering 12, 1097

    work page 2025

  41. [41]

    Theory and applications of abstract semilinear Cauchy problems

    Magal, P., Ruan, S., et al., 2018. Theory and applications of abstract semilinear Cauchy problems. volume 201. Springer

    work page 2018

  42. [42]

    The dynamics of physiologically structured populations

    Metz, J.A., Diekmann, O., 2014. The dynamics of physiologically structured populations. Springer

    work page 2014

  43. [43]

    Mathematical Biology: I

    Murray, J.D., 2007. Mathematical Biology: I. An Introduction. volume 17. Springer Science & Business Media

    work page 2007

  44. [44]

    Analysis of a nonlinear age-structured sis epidemic model with spatial diffusion: S

    Nag, S., Tumuluri, S.K., 2025. Analysis of a nonlinear age-structured sis epidemic model with spatial diffusion: S. nag, sk tumuluri. Computational and Applied Mathematics 44, 250

    work page 2025

  45. [45]

    Optimal control of a size-structured unicellular algae model in harvesting and sewage treatment

    Ni, D., Ma, W., 2023. Optimal control of a size-structured unicellular algae model in harvesting and sewage treatment. Computers & Mathematics with Applications 145, 240–259

    work page 2023

  46. [46]

    Semigroups of linear operators and applications to partial differential equations

    Pazy, A., 2012. Semigroups of linear operators and applications to partial differential equations. Springer Science & Business Media

    work page 2012

  47. [47]

    Transport equations in biology

    Perthame, B., 2007. Transport equations in biology. Springer

    work page 2007

  48. [48]

    A gentle introduction to physiologically structured population models, in: Structured-population models in marine, terrestrial, and freshwater systems

    de Roos, A.M., 1997. A gentle introduction to physiologically structured population models, in: Structured-population models in marine, terrestrial, and freshwater systems. Springer, pp. 119–204

    work page 1997

  49. [49]

    Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation

    Scarabel, F., Breda, D., Diekmann, O., Gyllenberg, M., Vermiglio, R., 2021. Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation. Vietnam Journal of Mathematics 49, 37–67

    work page 2021

  50. [50]

    Compact sets in the space𝑙𝑝(𝑜, 𝑡;𝑏)

    Simon, J., 1986. Compact sets in the space𝑙𝑝(𝑜, 𝑡;𝑏). Annali di Matematica pura ed applicata 146, 65–96. J. Yu and L. S. Wang:Preprint submitted to ElsevierPage 24 of 25 Bilinear control of age–space structured populations

    work page 1986

  51. [51]

    Effects of size-and sex-selective harvesting: An integral projection model approach

    Stubberud, M.W., Vindenes, Y., Vøllestad, L.A., Winfield, I.J., Stenseth, N.C., Langangen, Ø., 2019. Effects of size-and sex-selective harvesting: An integral projection model approach. Ecology and Evolution 9, 12556–12570

    work page 2019

  52. [52]

    Mathematics in population biology

    Thieme, H.R., 2018. Mathematics in population biology. Princeton University Press

    work page 2018

  53. [53]

    Optimalcontrolofpartialdifferentialequations:theory,methods,andapplications.volume112

    Tröltzsch,F.,2010. Optimalcontrolofpartialdifferentialequations:theory,methods,andapplications.volume112. AmericanMathematical Soc

    work page 2010

  54. [54]

    Analysis framework for stochastic predator–prey model with demographic noise

    Wang, L.S., Yu, J., 2025. Analysis framework for stochastic predator–prey model with demographic noise. Results in Applied Mathematics 27, 100621

    work page 2025

  55. [55]

    Algebraic–spectralthresholdsanddiscrete–continuousstabilitytransferinleslie–gowersystems

    Wang,L.S.,Yu,J.,2026. Algebraic–spectralthresholdsanddiscrete–continuousstabilitytransferinleslie–gowersystems. ElectronicResearch Archive 34, 251–290

    work page 2026

  56. [56]

    Analysis and mean-field limit of a hybrid pde-abm modeling angiogenesis-regulated resistance evolution

    Wang, L.S., Yu, J., Li, S., Liu, Z., 2025a. Analysis and mean-field limit of a hybrid pde-abm modeling angiogenesis-regulated resistance evolution. Mathematics 13, 2898

    work page

  57. [57]

    The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems

    Wang, L.S., Yu, J., Liang, Y., Zhang, J., 2026a. The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems. Electronic Research Archive 34, 4248–4289

    work page

  58. [58]

    Elliptic criticality versus volterra memory in indirect chemotaxis cascades

    Wang, L.S., Yu, J., Liang, Y., Zhang, J., 2026b. Elliptic criticality versus volterra memory in indirect chemotaxis cascades. Transport Phenomena 1, 20260061

    work page

  59. [59]

    A damage-structured pde model of stem cell hierarchies: The dual role of dedifferentiation in tissue homeostasis and aging

    Wang, L.S., Yu, J., Liu, Z., 2026c. A damage-structured pde model of stem cell hierarchies: The dual role of dedifferentiation in tissue homeostasis and aging. Plos one 21, e0335163

    work page

  60. [60]

    Multi-strategy hybrid improved intelligent algorithm for solving uav-mtsp

    Wang, Z., Wang, D., Yu, J., 2025b. Multi-strategy hybrid improved intelligent algorithm for solving uav-mtsp. Information Technology and Control 54, 413–438

    work page

  61. [61]

    Populationmodelsstructuredbyage,size,andspatialposition,in:Structuredpopulationmodelsinbiologyandepidemiology

    Webb,G.,2008. Populationmodelsstructuredbyage,size,andspatialposition,in:Structuredpopulationmodelsinbiologyandepidemiology. Springer, pp. 1–49

    work page 2008

  62. [62]

    Theory of nonlinear age-dependent population dynamics

    Webb, G.F., 1985. Theory of nonlinear age-dependent population dynamics. CRC Press

    work page 1985

  63. [63]

    Anage-structuredsyphilismodel,ii:optimalcontrolandnumericalsimulation,in:ProceedingsA,TheRoyalSociety

    Wu,P.,Ruan,S.,2025. Anage-structuredsyphilismodel,ii:optimalcontrolandnumericalsimulation,in:ProceedingsA,TheRoyalSociety. p. 20230809

    work page 2025

  64. [64]

    Analysis and optimal control of a hierarchical age-structured population model in a polluted environment

    Wu, Z., Luo, Z., Zhang, T., 2025. Analysis and optimal control of a hierarchical age-structured population model in a polluted environment. Journal of Mathematical Analysis and Applications 542, 128802

    work page 2025

  65. [65]

    Beyonddiagonalnoise:Abetterpredator-preymodelingframeworkwithcross-covariance

    Yu,J.,Wang,L.S.,2026. Beyonddiagonalnoise:Abetterpredator-preymodelingframeworkwithcross-covariance. PLoSOne21,e0350127

    work page 2026

  66. [66]

    From microscopic damage to macroscopic games: a dimensionality reduction of stem cell homeostasis

    Yu, J., Wang, L.S., Ban, S., Liang, Y., 2026a. From microscopic damage to macroscopic games: a dimensionality reduction of stem cell homeostasis. Transport Phenomena 1, 20260037

    work page

  67. [67]

    Rigorous analysis of a nonlocal transport–renewal system for physiologically structured populations

    Yu, J., Wang, L.S., Liang, Y., 2026b. Rigorous analysis of a nonlocal transport–renewal system for physiologically structured populations. Mathematical Methods in the Applied Sciences

    work page

  68. [68]

    Pattern suppression and recovery under one-way versus two-way chemotactic coupling in hybrid partial differential equation–ordinary differential equation models

    Yu, J., Wang, L.S., Liu, Z., Liu, J., 2026c. Pattern suppression and recovery under one-way versus two-way chemotactic coupling in hybrid partial differential equation–ordinary differential equation models. Transport Phenomena . J. Yu and L. S. Wang:Preprint submitted to ElsevierPage 25 of 25

    work page