Recovering the spatiotemporally dependent diffusion and advection coefficients in a pathway-based Keller--Segel model
Abstract
We investigate an inverse coefficient problem for an augmented Keller--Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient D(x,t) and the scalar advection coefficient K(x,t) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.
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BibTeX
@misc{li2026recovering,
title = {{Recovering the spatiotemporally dependent diffusion and advection coefficients in a pathway-based Keller--Segel model}},
author = {Yu'an Li and Lingyun Qiu and Min Tang and Hui Yu and Shenwen Yu and Siqin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23364},
eprint = {2609.23364},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23364},
}
RIS
TY - UNPB
TI - Recovering the spatiotemporally dependent diffusion and advection coefficients in a pathway-based Keller--Segel model
AU - Yu'an Li
AU - Lingyun Qiu
AU - Min Tang
AU - Hui Yu
AU - Shenwen Yu
AU - Siqin Zheng
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23364
UR - https://arxiv.org/abs/2609.23364
AB - We investigate an inverse coefficient problem for an augmented Keller--Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient D(x,t) and the scalar advection coefficient K(x,t) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.
ER -