Attraction-Repulsion Chemotaxis System

• Attraction-repulsion chemotaxis systems are mathematical models describing cell migration influenced by simultaneous chemoattractant and chemorepellent cues using nonlinear, cross-coupled PDEs.
• They integrate logistic population regulation, nonlinear diffusion, and detailed energy estimates to capture phenomena such as traveling waves, blow-up prevention, and steady-state patterns.
• Analytical and numerical methods, including functional inequalities and spectral analysis, provide actionable insights into stability, boundedness, and biological interpretation.

An attraction-repulsion chemotaxis system describes the migration of biological agents (typically cells or bacteria) in response to simultaneous attractive and repulsive chemical cues. These systems exhibit nonlinear, cross-coupled partial differential equations (PDEs) modeling how cell density evolves under the influence of spatial gradients of both chemoattractants and chemorepellents, often complemented by logistic population regulation and signal kinetics. The mathematical analysis focuses on topics including global existence, blow-up prevention, traveling wave solutions, regularity, and steady-state stability under varying magnitudes and nonlinearities of the chemotactic responses.

1. Mathematical Formulation and Representative Models

The canonical form of an attraction-repulsion chemotaxis system involves a parabolic PDE for the cell density $u(x,t)$ coupled with equations for the chemoattractant $v_1(x,t)$ and chemorepellent $v_2(x,t)$ concentrations:

$$\begin{cases} u_t = \Delta u - \chi_1 \nabla \cdot (u \nabla v_1) + \chi_2 \nabla \cdot (u \nabla v_2) + u(a - b u), \ \tau v_{1,t} = (\Delta - \lambda_1 I)v_1 + \mu_1 u, \ \tau v_{2,t} = (\Delta - \lambda_2 I)v_2 + \mu_2 u, \end{cases} \quad x \in \mathbb{R}^N, \, t > 0$$

with constants $\chi_i$, $\mu_i$, $\lambda_i>0$, population growth rates $a>0$, $b>0$, and possibly $\tau$ indicating the nature (parabolic or elliptic) of signaling kinetics (Salako, 2018).

This structure generalizes to bounded domains $\Omega \subset \mathbb{R}^n$ with Neumann boundary conditions and may incorporate terms modeling nonlinear diffusion, density-dependent sensitivities, nonlocal logistic sources, and various forms of chemical production and consumption (Shanmugasundaram et al., 11 Jul 2025, Li et al., 2022, Frassu et al., 2021).

2. Existence, Boundedness, and Blow-Up Criteria

Global Existence and Uniform Boundedness

For classical Keller–Segel models, solutions can exhibit finite-time blow-up due to strong aggregation. However, the following mechanisms have been rigorously shown to ensure global existence and boundedness in attraction-repulsion systems:

• Dominant Repulsion: If the repulsion flux parameter(s) or chemical production rates outweigh the attraction, collapse is suppressed and solutions remain bounded (e.g., $\chi\alpha - \xi\gamma < 0$) (Chiyo et al., 2021, Wakui et al., 22 Nov 2025, Lankeit, 2021).
• Sublinear/Saturated Signal Production: For sublinear signal production (e.g., $v_t = \Delta v + \alpha u^p - \beta v$ with $0 < p < 1$), aggregation is limited regardless of the relative sizes of attraction/repulsion parameters (Pintus et al., 2019, Shanmugasundaram et al., 11 Jul 2025, Frassu et al., 2021).
• Nonlocal Logistic Regulation: Nonlocal terms such as $\mu u^m(1 - \int_\Omega u)$ (with $m > 1$ and suitable powers for chemotactic fluxes) regulate the total cell mass and provide essential damping (Shanmugasundaram et al., 11 Jul 2025).
• Double Saturation: Simultaneous consumption of both chemical cues by the cells prevents runaway signal amplification and mass clustering (Frassu et al., 2021).
• Nonlinear Diffusion and Gradient Damping: Algebraic (e.g., $(u+1)^{m-1}\nabla u$) or dissipative gradient terms $-c|\nabla u|^\gamma$ (with $\gamma$ above a model-dependent threshold) have been shown to preclude $\delta$-type singularity formation (Li et al., 6 May 2024).

Blow-Up and its Prevention

Blow-up typically occurs for attraction-dominated regimes:

• If $\chi\alpha-\xi\gamma>0$, solutions in three or higher dimensions can blow up in finite time for suitable initial data, even under fully parabolic dynamics (Lankeit, 2021, Chiyo et al., 2021).
• In regimes with only logistic damping and $k>1$ only slightly larger than 1 in $-\mu u^k$ terms, blow-up may still occur unless the damping is sufficiently strong (Chiyo et al., 2021).

Explicit lower bounds for the blow-up time have been derived and are directly dependent on the magnitude of initial concentrations and the interplay between chemoattractive and chemorepulsive parameters (Viglialoro, 2019, Le et al., 2022).

3. Traveling Waves and Their Admissible Speeds

Traveling wave solutions are studied via ansatz $u(x,t) = U(x-ct)$, accompanied by appropriate reductions to autonomous ODE systems. For full parabolic attraction-repulsion systems with logistic sources, robust analytical frameworks yield:

• A range of admissible wave speeds $c^* < c < c^{**}$, explicitly determined by system parameters and convolution integrals encoding chemotactic effects (Salako, 2018).
• Sharp decay properties: $\lim_{z\to \infty} U(z)/e^{-\mu z} = 1$ for $\mu$ associated with the traveling wave speed $c_\mu$.
• Limiting behavior: As chemotactic sensitivities $(\chi_1,\chi_2) \to 0^+$, the maximal admissible wave speed diverges while the minimal approaches the Fisher-KPP speed, reflecting recovery of classical population dynamics (Salako, 2018).
• No traveling wave solutions exist for speed $c < 2\sqrt{a}$, corresponding to a critical threshold extracted via linearization and spectral techniques.

4. Functional Inequalities and A Priori Estimates

Analysis of attraction-repulsion systems employs sophisticated estimates:

• Weighted energy functionals with exponential (or power-law) weights reflecting chemical concentrations and cell density (Frassu et al., 2021, Columbu et al., 2022).
• Gagliardo–Nirenberg, Ehrling, and Moser–Alikakos inequalities connect higher moments and gradient integrals to control global norms and close a priori bounds (Frassu et al., 2020, Chiyo et al., 2021).
• Differential inequalities of the form $\frac{d}{dt} y(t) + A y^\delta \le B$ with $\delta > 1$ guarantee uniform boundedness, provided the nonlinearities are appropriately matched (Li et al., 2022, Chiyo et al., 2021).
• Negative cross-diffusion (repulsive) coupling terms enable absorption of otherwise blow-up driving chemotactic aggregation terms into dissipative structures (Chiyo et al., 2021, Columbu et al., 2022).
• Explicit lower bounds on blow-up time are attained via careful interpolation inequalities, spectral bounds on chemical signals, and integrated energy dissipation (Viglialoro, 2019, Le et al., 2022).

5. Stability of Steady States and Patterns

Stability analysis for constant equilibrium states in the whole space reveals critical thresholds:

• Purely attractive systems admit stable steady states for $A < \lambda_1/\beta_1$ (Wakui et al., 22 Nov 2025).
• Purely repulsive systems have all constant steady states stable.
• For composite attraction-repulsion, boundaries $A < A_*(\beta_j,\lambda_j)$ partition stability regions in the parameter space, dictated by the comparative ratios of attraction and repulsion strengths and decay rates (Wakui et al., 22 Nov 2025).
• Linearization and Fourier analysis of the nonlocal operator yield explicit spectral criteria—decay of perturbations and asymptotic stability are inherited from subcritical regimes.

6. Extensions, Numerical Validation, and Biological Interpretation

Modern works extend classical models by:

• Incorporating nonlinear diffusion, signal-dependent sensitivities, quasilinear kinetics, and coupling to incompressible fluid flow (Duarte-Rodríguez et al., 2017).
• Numerical simulations validate critical thresholds for global existence and finite-time blow-up, track $\delta$-peak formation, and allow sweeping across production/consumption parameters for comprehensive phase diagrams (Columbu et al., 2022, Shanmugasundaram et al., 11 Jul 2025).
• Biologically, systems balancing attraction and repulsion with consumption/production of cues explain stabilization of cell distributions, suppression of pathological aggregates, and robust regularization even from highly irregular (measure-valued) initial data (Heihoff, 2022, Frassu et al., 2021, Viglialoro, 2019).

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In summary, attraction-repulsion chemotaxis systems generalize classic Keller–Segel-type models with dual chemotactic cues and demonstrate a rich set of mathematical phenomena—boundedness, pattern formation, singularity development, and traveling waves—with dynamical regimes sharply characterized by parameter thresholds, functional inequalities, and dissipative mechanisms. Analytical and numerical progress continually clarifies the impact of nonlocal regulation, nonlinearities, and the interplay of aggregation and dispersion in cell migration and population dynamics.