arXiv subjects

Louis Shuo Wang

Publications and source records attributed to Louis Shuo Wang.

At least 19 recordsLinked to original sources

arXiv2026

Size-Selective Threshold Harvesting under Nonlocal Crowding and Exogenous Recruitment

We propose a nonlinear size-structured fishery model for externally recruited stocks under size-selective harvesting. The population density satisfies a McKendrick--von Foerster transport equation in which growth and natural mortality depend on a nonlocal crowding index, while harvesting acts as a bounded size-dependent mortality control. Unlike standard self-recruiting formulations, recruitment is prescribed as a lower-boundary inflow, making the model suitable for enhancement fisheries or analyses conditional on juvenile input. For the no-harvest baseline, we derive the stationary size profile and reduce the nonlinear equilibrium problem to a scalar closure equation, proving existence and uniqueness under a net monotonicity condition. We introduce an intrinsic replacement index and show why, in this externally forced setting, it is a viability diagnostic rather than a persistence threshold. A formal state--adjoint system yields a bang--bang switching rule; under weak coupling and single crossing, the optimal policy has a threshold structure. Numerical experiments validate the approximation and sensitivity trends

math.OC↗

arXiv2026

A Nonlocal Perfusion-Gated Angiogenesis System: Global Weak Solutions, Fast-Signal Limits, and Invasion Fronts

Continuum angiogenesis systems often use local endothelial or vessel density as an oxygen-delivery proxy without distinguishing structurally formed vessels from pressure-supported vascular function. We formulate a two-dimensional model in which lumenized density determines a normalized nonlocal conductivity, a globally solved pressure field, and a flow-functional density that gates oxygen delivery and vessel regression. For fixed positive regularization parameters and signal relaxation times, we prove Lipschitz stability of the pressure--perfusion map and global bounded weak solvability; strong vessel compactness follows from a nonlocal ordinary differential equation stability estimate rather than spatial smoothing. As the oxygen and vascular endothelial growth factor timescales vanish simultaneously, weak solutions converge along a subsequence to a parabolic--elliptic--ordinary differential equation system, with both fast fields converging strongly in $L^2(0,T;H^1)$. A locally frozen one-dimensional reduction admits monotone fronts precisely at or above the threshold $B+2\sqrt{DR}$ for $B\geq-\sqrt{DR}$ and $-DR/B$ otherwise. Finite-volume experiments verify the analytical mechanisms and show that equal-mass vessel fields can generate distinct functional masses and oxygenation levels.

math.AP↗

arXiv2026

Smooth expanding planar simple waves for Euler--Poisson--Boltzmann: uniform stability and quasineutral expansion

We study the warm-ion Euler--Poisson system with Maxwell--Boltzmann electrons on the cylinder $\R\times\T$ in the quasineutral regime. Taking a smooth expanding planar simple wave of the effective Euler system as the reference state, we construct an even Debye expansion through arbitrary finite order $M$ with a residual of $O(\eps^{2M+2})$. We rigorously establish nonlinear stability on every fixed interval $[t_0,T]$ ($t_0\ge0$), achieving a lifespan and energy constants strictly independent of the Debye length $0<\eps\le\eps_0$. To overcome the singular scaling of the electric field, we develop a novel compensated energy topology that couples the warm-ion symmetrizer to the time-differentiated nonlinear Poisson constraint. By integrating the top-order electric work directly into the time derivative of a weighted energy functional, this mechanism controls the potential in $H^s$ and its gradient in $\eps H^s$, completely eliminating the $\eps^{-1}$ loss typically encountered in the momentum equation. This framework successfully governs distinct neutral end states, captures genuinely two-dimensional rotational perturbations, and yields an arbitrary-order quasineutral asymptotic expansion for prepared data, providing a critical analytical foundation for the geometric theory of multidimensional quasineutral rarefactions.

math.AP↗

arXiv2026

Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff $G(x_1,x_2)=x_1\veeαx_2$. The main novelty is a measure-valued variational formulation: the stopping gain $Γ=c+rG-\mathcal LG$ is shown to be a signed Radon measure whose singular component is supported on the kink diagonal $\{x_1=αx_2\}$, and this component is computed explicitly. We prove that the value admits the killed-resolvent representation \[ V=G-R_r^{\mathcal C}Γ, \] where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.

math.OC↗

arXiv2026

Endogenous Feedback in Size-Structured Transport Equations

We study a nonlinear size-structured transport equation where the endogenous scalar output $E(t)=\int_{l_0}^{l_m}χ(l)x(t,l)\,dl$ feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point $E=\mathcal K(E)$. Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation $E=Φ(E)$. We prove uniqueness below the sharp margin $1-Φ'(E)>0$ and identify $Φ'(E)=1$ as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation $\mathcal E(λ)=1$, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity $\mathcal E(0)=Φ'(E^*)=B(0)$, linking closure resonance, spectral crossing, and adjoint loop gain.

math.AP↗

arXiv2026

Chemotactic Feedback Controls Patterning in Hybrid Tumor--Stroma Model

Motivated by an ongoing collaboration with clinical oncologists and pathologists, we develop a hybrid partial differential equation--ordinary differential equation (PDE--ODE) framework that captures (i) competition between susceptible and resistant phenotypes, (ii) stromal state switching, and (iii) a clinically realistic open-loop, single-dose therapeutic agent $I$ with diffusion and clearance. Clinical management of solid tumors is increasingly limited by spatial heterogeneity and therapy-induced resistance niches that are difficult to predict from well-mixed models. We establish a rigorous mathematical backbone with forward invariance of the nonnegative cone and global-in-time well-posedness. Exploiting the decoupled drug equation $\partial_t I=d_IΔI-γ_I I$, we prove a long-time reduction during washout and show that the damped base dynamics admit no diffusion-driven (Turing-type) instability. We then formulate a directionality--damping principle: unidirectional (open-loop) sensing yields at most transient focusing, whereas bidirectional (closed-loop) feedback reshapes the effective mobility and produces explicit thresholds separating stable homogeneity, finite-band patterning (resistance niche formation), and aggregation when strong parabolicity is violated. Reproducible simulations corroborate this classification and highlight when flux regularization is required for physical realism.

math.AP↗

arXiv2026

Age-Structured Harvesting Models: A Structural Comparison of Rate-Control and Effort-Control Optimality Systems

We study optimal harvesting in continuous-time, age-structured population models of McKendrick--von Foerster type, and we compare two harvesting mechanisms. In the \emph{rate-control} formulation, harvesting enters the state equation as an additive removal term; in the \emph{effort-control} formulation, harvesting acts multiplicatively as an additional mortality intensity and the mortality coefficient depends on the aggregate stock. For the rate-control problem we first establish existence of an optimal control for the infinite-horizon discounted problem, and we then derive, \emph{under explicitly stated regularity and constraint-qualification assumptions}, a conditional Pontryagin-type necessary optimality system consisting of the adjoint equation, the terminal-age and transversality conditions, the switching relations for the distributed and boundary controls, and the complementary-slackness relation for the state constraint. For the effort-control problem we \emph{formally} derive the associated adjoint equation and identify the nonlocal coupling term generated by aggregate (density) dependence, with the sign of that term verified by a step-by-step variational computation; a rigorous infinite-horizon maximum principle for this nonlinear, nonlocally coupled problem is beyond the present scope and is stated as such. We complement the analysis with autonomous stationary reductions, with explicit representations of the state and adjoint, and with a reproducible numerical study. The central message is structural: the harvesting mechanism is not a cosmetic modelling choice. Rate-control produces an additive/affine/local optimality structure, whereas effort-control produces a multiplicative/nonlinear/nonlocal one, with direct consequences for persistence, stationary profiles, and bioeconomic interpretation.

math.OC↗

arXiv2026

Bilinear control of age--space structured populations

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 Fréchet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation $\ell_{\bar y,\bar u}(p)(t)χ(a,x)$; 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.

math.OC↗

arXiv2026

Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

We develop a reaction--diffusion--chemotaxis model for spatial tumour--immune--chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies $σ_0/δ>1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity $ξ_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

math.AP↗

arXiv2026

FTU-Seek: Foundation Model-Guided Hard-Negative Learning for Sparse Functional Tissue Unit Segmentation

Functional tissue units (FTUs), including tertiary lymphoid structures (TLSs), blood vessels, and glands, encode localized immune, vascular, and epithelial organization in histopathology. Accurate quantification of these structures is important for studying tissue architecture and disease-associated tissue organization. However, FTUs are frequently sparse, heterogeneous, and surrounded by large amounts of morphologically similar background tissue, making automated segmentation in whole-slide images (WSIs) challenging. We therefore developed FTU-Seek, a pathology foundation model-guided framework that treats morphology-aware negative-patch selection as a key component of sparse FTU segmentation. FTU-Seek uses frozen multi-depth features from the UNI pathology foundation model to train a patch-level classifier that distinguishes FTU-containing from FTU-absent tissue. Target-absent patches are subsequently ranked according to their predicted target-containing probabilities, and the highest-scoring hard negatives are selected through a static Top$K$ strategy to construct compact segmentation training sets. The framework was evaluated using five-fold cross-validation and internal test cohorts across TLS, blood-vessel, and gland segmentation tasks, with an additional independent 30-WSI held-out cohort for TLS. Positive-only, all-tissue, random-negative, and matched random Top$K$ sampling strategies served as comparators. Segmentation-derived phenotypes were further explored in external TCGA cohorts.

cs.CV↗

arXiv2026

Rigorous Analysis of a Nonlocal Transport--Renewal System for Physiologically Structured Populations

We develop a rigorous analytical framework for a class of physiologically structured population models with two internal state variables, nonlocal ecological feedbacks, dynamic resources, inter-zone transfer, and selective harvesting. The full model is a coupled nonlinear PDE--ODE transport--renewal system with endogenous inflow at the recruitment boundary, a setting in which transport, nonlocal dependence, and boundary renewal interact at the same level. For this full nonautonomous multi-zone system, we prove finite-horizon well-posedness in a positive $L^{1}$-based state space, including global existence on arbitrary bounded time intervals, uniqueness, nonnegativity, and continuous dependence on initial data, environmental forcing, and harvesting effort. We then isolate an autonomous single-zone reduction at extinction and construct a positive compact next-generation operator on the recruit space. In a further nonlinear stationary reduction, we prove that supercriticality of the basic reproduction number $\mathcal R_{0}>1$ yields existence of a nontrivial stationary state under a parametrized compact-operator hypothesis encoding density-dependent renewal feedback. Finally, for a finite-horizon harvest objective over a compact Lipschitz-regular admissible class, we establish existence of an optimal control. The results separate what can be proved for the full climate-explicit system from what can be justified only after autonomous reduction, thereby clarifying the mathematical scope of threshold and control theory for structured populations.

math.AP↗

arXiv2026

A transport-only null model for apparent heterogeneity in diffusively dosed organoid arrays

Spatial transport can create apparent biological heterogeneity even when organoids are intrinsically identical. We develop a transport-to-phenotype null model for diffusively dosed liver-cancer organoid arrays. The model couples bulk diffusion and clearance to partially accessible adsorption, reversible surface residence, productive internalization, and intracellular state dynamics. Matched asymptotics reduce the perforated-domain problem to a Green-function system, while renewal resolvents describe desorption, re-adsorption, and residence-time effects. Across $2000$ random ten-organoid arrays with localized dosing, the predicted transport-only maturation coefficient of variation has median $0.623$; one-factor design changes move this median between $0.27$ and $0.86$. After matching array-mean exposure, distributed dosing reduces the baseline spread approximately fivefold. The analysis also shows that, in a conservative reflecting chamber, desorption changes uptake timing and allocation but not total eventual uptake; reductions in total uptake require a competing loss channel. Residence laws with equal means can nevertheless produce different transient phenotypes. The spatial reduction is verified against finite-element solutions of the full PDE, and the time reconstruction against numerical Laplace inversion. Finally, a large-batch theorem shows that increasing batch size averages independent process variation but not shared line or batch effects. The framework provides a geometry-specific null against which measured organoid heterogeneity can be assessed.

math.AP↗

arXiv2026

From Flows to Maps: Sampling Laws for Attractor Intensity and Bounded-Noise Escape

Intensity of attraction quantifies the largest amplitude of a persistent bounded disturbance that an attractor can withstand without loss of controlled confinement in its basin. Although intensity has been formulated separately for flows and maps, its behavior under temporal sampling has remained unresolved. We establish an explicit correspondence between the intensity $μ(A)$ of a continuous-time attractor and the intensity $μ_h(A)$ of its exact time-$h$ map. For an $L$-Lipschitz vector field, \[ \frac{μ(A)}{1+Lh} \leq \frac{μ_h(A)}{h} \leq μ(A)\frac{e^{Lh}-1}{Lh}, \] and hence $μ_h(A)/h\toμ(A)$. The resulting first-order rate is sharp in general, while smooth scalar escape geometries can exhibit second-order convergence. We extend the framework to one-step numerical methods through a stability theory for block intensity and to attracting invariant graphs over compact invertible nonautonomous bases, obtaining uniform sampling convergence over the forcing phase. For bounded-support random perturbations, normalized discrete intensity is identified with the pathwise safety threshold; above it, finite escape follows under an explicit finite-exit condition, while escape probabilities require additional assumptions on the noise law. We also show that the discrete state--normal boundary map converges to the normalized Pontryagin boundary system governing extremal reachable-set boundaries. Exact scalar benchmarks, a grazing resilience model, planar Duffing escape, anisotropic disturbances, periodic and quasiperiodic forcing, and transfer-operator computations illustrate the theory. These results give intensity estimated from discrete observations or simulations a sampling-independent continuous-time meaning.

math.DS↗

arXiv2026

Optimal Harvesting of Size-Structured Populations with Environmental Feedback and Fixed Recruitment Flux

We study a nonlinear size-structured transport model with distributed harvesting and prescribed recruitment flux, where environmental feedback is determined by a scalar population functional. After establishing global well-posedness on $L^1$, we reduce stationary equilibria to a scalar closure equation. This reduction reveals that loss of equilibrium uniqueness occurs through a generic fold, mathematically characterizing critical population transitions. On uniformly nonresonant equilibrium branches, we prove the existence of optimal stationary harvesting policies via the direct method of the calculus of variations. We then derive a boundary-corrected adjoint equation and establish an identity equating equilibrium sensitivity with the adjoint loop gain. This relation yields explicit criteria for the persistence and creation of optimal harvesting thresholds. Collectively, these results provide a unified analytical framework connecting environmental feedback, equilibrium structure, and optimal harvesting.

math.AP↗

arXiv2026

Fredholm--residue selection of the unsteady Kutta amplitude

We give an operator-theoretic interpretation of unsteady Kutta selection in trailing-edge acoustic receptivity. The inviscid acoustic--wake problem leaves one outgoing wake amplitude undetermined. We show that, under explicit structural hypotheses, this amplitude is the same scalar obtained from three representations: cancellation of the inverse-square-root edge singularity, Fredholm compatibility of the viscous lower-deck problem, and the residue of the Kutta-normalized transform solution at the downstream wake pole: $\displaystyle A = -\frac{C_-^{(0)}}{C_-^{(KH)}} = -\frac{\langle \mathbf F_{\rm inc},Ψ^\ast\rangle}{\langle \mathbf F_{KH},Ψ^\ast\rangle} = i\operatorname*{Res}_{α=α_{KH}}\mathcal M(α)$. The inner Fredholm--edge mechanism is verified exactly in a linear-shear lower-deck model, where the primal shear and adjoint velocity are Airy fields and the edge concomitant is nonzero outside a discrete resonance set.

math.NA↗

arXiv2026

Vital-Rate Feedback and Threshold Harvesting in Size-Structured Populations

Size-selective harvesting is often justified by the intuition that larger individuals have higher value and should therefore be harvested above a critical size. We study a controlled size-structured transport model in which a scalar environmental variable, generated by the population itself, modifies the vital rates $g(E,l)$ and $μ(E,l)$. The first result is a corrected stationary closure theory: crowding-suppressed growth increases residence density at the inflow, so the closure derivative is not determined by pointwise profile monotonicity. Instead, $Φ'(E)=\mathsf A(E)-\mathsf C(E)$, an integrated balance between residence-time amplification and cumulative survival loss. The second result is an exact stationary adjoint reduction. The nonlocal switching correction has rank one, $S=S_{\rm red}-\frac{A}{1-B}ψ$, and its zero-discount feedback gain satisfies the identity $B(0)=Φ'(E^*)$. Thus the same scalar governs stationary closure sensitivity and threshold fragility. We also establish finite-horizon well-posedness and compactified optimal-control existence in a spatial-$BV$ policy class. Numerical certification in a density-dependent von Bertalanffy model shows when minimum-size harvesting persists and when vital-rate feedback creates multiple-switch harvest windows.

math.AP↗

arXiv2026

A Measure-Valued Obstacle Problem for an Obliquely Reflected Diffusion with a Max-Type Payoff

We study an obliquely reflected optimal stopping problem in the nonnegative quadrant with nonsmooth max-type payoff $G(x)=x_1\veeαx_2$, and we develop a measure-valued potential-theoretic formulation of the associated obstacle problem. The kink of $G$ on the diagonal $x_1=αx_2$ produces a singular surface measure in the distributional generator, while the oblique reflection directions generate boundary local-time contributions on the coordinate faces. Together with the absolutely continuous stopping gain, these terms define a total signed stopping measure $\Gtot$. We derive the corresponding reflected Itô--Tanaka identity, prove a killed-resolvent representation of the value function in the continuation region, and show that the unrestricted reflected resolvent is generally incorrect because the process is not absorbed on the stopping set. The free boundary is formulated through a continuation-side trace condition for the killed potential. Under a vertical monotonicity hypothesis on $V-G$, the stopping set is shown to have an epigraph form. We finally prove a verification theorem: any admissible epigraph candidate satisfying contact, strict continuation, reflected Neumann compatibility, growth, the trace condition, and measure-superharmonicity coincides with the value function, and its first entry time is optimal.

math.AP↗

arXiv2026

Killed resolvents and measure-valued stopping gains for reflected optimal stopping with max-type rewards

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the positive quadrant with nonsmooth max-type reward $G(x_1,x_2)=x_1\vee αx_2$. The paper develops a conditional measure-theoretic framework for the associated reflected obstacle problem. The main innovation is to show that the stopping gain $Γ=c+rG-\mathcal LG$ is a signed measure, not a function: the kink of $G$ generates an explicit negative surface measure on $Δ=\{x_1=αx_2\}$. We then prove that the correct potential representation uses the resolvent of the reflected diffusion killed on first entry into the stopping set, rather than the unrestricted reflected resolvent. Under explicit monotonicity, regularity, and measure-superharmonicity assumptions, we derive an epigraph representation, a continuation-side boundary-trace condition, and a candidate verification theorem. The framework clarifies hidden regularity and uniqueness assumptions in multidimensional nonsmooth optimal stopping.

math.PR↗