arXiv subjects

Federico Herrero-Hervás

Publications and source records attributed to Federico Herrero-Hervás.

8 recordsLinked to original sources

arXiv2026

Transient instability and recurrent aggregation in a two-species chemotaxis-competition system under environmental periodicity

This work investigates recurrent pattern formation in a two--species chemotaxis system with competitive dynamics under the assumption that the carrying capacities of both species are periodic in time. By considering a globally attracting periodic solution to the associated nonautonomous ODE system as a spatially homogeneous base state, we derive sufficient conditions for the existence of a critical chemotactic sensitivity of the first species, denoted by $χ_{\text{crit}}$. Whenever this threshold is exceeded, the periodic state becomes linearly unstable over a nonempty set of times within each environmental cycle. This gives rise to a recurrent alternation of unstable and stable intervals, during which spatial perturbations are respectively amplified and damped. The analytical derivation is complemented with numerical simulations, in both periodic coexistence and competitive exclusion regimes. Finally, we discuss the temporal location of the instability sets and the possibility of negative instability thresholds.

math.AP↗

arXiv2026

Global stability in a negative chemotaxis system with chemically induced lethality

In this paper, we investigate the long-time dynamics of a repulsive Keller-Segel chemotaxis system. The model features negative chemotaxis, logistic growth and a cell death term, accounting for a lethal chemorepellent that is self-produced by the cells and externally supplied. We prove that, for constant chemorepellent supplies, depending on their magnitude with respect to the logistic growth rate, solutions converge in $L^\infty$ norm toward extinction of the population, or equilibrate toward a nontrivial spatially homogeneous steady state.

math.AP↗

arXiv2026

Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system

This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $Ω\subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0 \in W^{1,\infty}(Ω)$ and $f \in C^1\big(\barΩ \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma.

math.AP↗

arXiv2025

Asymptotics and periodic dynamics in a negative chemotaxis system with cell lethality

This work studies the following system of parabolic partial differential equations \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = DΔu + χ\nabla \cdot(u \nabla v) + ru(1-u) - u v, \quad & x \in Ω, ~t > 0, \\ \displaystyle \frac{\partial v}{\partial t} = Δv + a u -v+ f(x,t), \quad & x \in Ω, ~t > 0, \end{cases} \end{equation*} modeling the negative chemotaxis interactions between a biological species and a lethal chemical substance that is supplied according to the known function $f(x,t)$. \\\\ It is shown that if $f$ converges to a spatially homogeneous function $\tilde{f}$ in a certain sense, then the solution $(u,v)$ satisfies $$ ||u-\tilde{u}||_{L^2(Ω)} + ||v-\tilde{v}||_{L^2(Ω)} \to 0 \quad \text{as } t \to \infty, $$ where $(\tilde{u},\tilde{v})$ is the solution to the associated ODE system \begin{equation*} \begin{cases} \displaystyle \frac{d \tilde{u}}{dt~} = r \tilde{u} (1 - \tilde{u}) - \tilde{u}\tilde{v}, \quad & t>0,\\ \displaystyle \frac{d \tilde{v}}{dt~} = a\tilde{u} - \tilde{v} + \tilde{f},\quad & t>0. \end{cases} \end{equation*} Some final remarks are given for the case in which $\tilde{f}$ is a time periodic function, and under which hypotheses do $(\tilde{u},\tilde{v})$ inherit this periodicity.

math.AP↗

arXiv2025

Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line

This work addresses the one-dimensional Cauchy problem for the doubly degenerate nutrient taxis model \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(u v u_x) - \frac{\partial}{\partial x}(u^2 v v_x) + u v, & x\in \mathbb{R}, ~t>0, \\ \displaystyle \frac{\partial v}{\partial t} = \frac{\partial^2 v}{\partial x^2} - u v, & x\in \mathbb{R}, ~t>0, \\ u(x,0) = u_0(x) \geq 0, \quad v(x,0) = v_0(x)>0, ~ & x\in \mathbb{R}, \end{cases} \end{equation*} which models pattern formation in bacterial populations. The global existence of weak solutions is established for initial data satisfying appropriate regularity and integrability conditions. To account for the degeneracy caused by $u_0$ not being strictly positive and the difficulties arising from the unboundedness of the domain, we consider a family of regularized problems posed on bounded intervals $(-\frac{1}{\varepsilon}, \frac{1}{\varepsilon})$, for $\varepsilon \in (0,1)$. Through adequate estimates uniform in $\varepsilon$, we construct global solutions to the system by passing to the limit using the Aubin-Lions lemma.

math.AP↗

arXiv2024

An explicit-implicit Generalized Finite Difference scheme for a parabolic-elliptic density-suppressed motility system

In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic-elliptic system modelling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit-implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.

math.NA↗

arXiv2023

A new coupled model for laser-induced thermal ablation accounting for tissue water vaporization

A new model is considered to describe the evolution of tissue temperature and water concentration during a thermal ablation process. The aim of this study is to expand the results obtained in Blauth et. al. 2020, where the tissue water concentration is considered as a known function, experimentally obtained. The model consists of two coupled equations: a parabolic partial differential equation for the temperature $θ$ and an ordinary differential equation for the water content $w$. The equation for $θ$ accounts for temperature diffusion, blood perfusion, the heat source provided by the laser applicator and the vaporization effects which act when the temperature approaches 100C. The equation for $w$ accounts for the loss of tissue water that occurs during vaporization. The model equations are solved in a cylindrical domain around the laser applicator and the results are compared with the existing experimental data for porcine liver tissue from Hübner et. al. 2017.

math.AP↗

arXiv2023

Convergence of a meshless numerical method for a chemotaxis system with density-suppressed motility

This article studies a parabolic-elliptic system modelling the pattern formation in E. coli bacteria in response to a chemoattractant known as acylhomoserine lactone concentration (AHL). The system takes into account certain bacterial strains with motility regulation, and the parameters of the equations represent the bacterial logistic growth, AHL diffusion and the rates of production and degradation of AHL. We consider the numerical solution to the system using the Generalized Finite Difference (GFD) Method, a meshless method known to effectively compute numerical solutions to nonlinear problems. The paper is organized to first explain the derivation of the explicit formulae of the method, followed by the study of the convergence of the explicit scheme. Then, several examples over regular and irregular meshes are given.

math.NA↗