[Paper Review] Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

Lianzhang Bao, Wenxian Shen|arXiv (Cornell University)|Dec 24, 2018

Mathematical Biology Tumor Growth33 references4 citations

TL;DR

This paper investigates the asymptotic dynamics of a logistic-type chemoattraction-repulsion system with fast-diffusing chemicals on the half-line, modeling invasive species spreading under chemotactic influence. It establishes that the population density converges uniformly to 1 under suitable conditions, demonstrating long-term stabilization despite nonlinear chemotaxis and time-dependent growth rates.

ABSTRACT

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $\mathbb{R}^+$, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

Motivation & Objective

• To understand the long-term behavior of a chemoattraction-repulsion system with logistic growth in a one-dimensional unbounded domain.
• To analyze the influence of chemotaxis toward attractants and away from repellents on population spreading and stabilization.
• To establish conditions under which the population density converges uniformly to a stable equilibrium value of 1.
• To provide a rigorous foundation for the free boundary problem by studying the limiting system on the half-line.
• To extend existing results on Keller-Segel type models to systems with both attraction and repulsion mechanisms.

Proposed method

• Formulates a parabolic-elliptic-elliptic system modeling population dynamics with chemoattraction (via $v_1$) and chemorepulsion (via $v_2$), where $v_1$ and $v_2$ are fast-diffusing chemicals.
• Imposes Neumann boundary conditions at $x=0$ and zero-flux conditions at the moving boundary in the free boundary case, with the half-line system serving as the formal limit.
• Uses comparison principles and upper/lower solutions to bound the population density $u$ via auxiliary ODEs driven by time-dependent coefficients.
• Applies iterative estimates to control the deviation of $v_1$ and $v_2$ from their equilibrium values, reducing the problem to a perturbed logistic equation.
• Employs a recursive argument based on $L^\infty$-norm decay of perturbations to show that $\|u(t,\cdot) - 1\|_{\infty} \to 0$ as $t \to \infty$.
• Relies on assumptions ensuring positivity of growth rates ($a_{\inf} > 0$, $b_{\inf} > 0$) and balance between attraction and repulsion ($b_{\inf} + \chi_2\mu_2 - \chi_1\mu_1 > 0$).

Experimental results

Research questions

• RQ1 Under what conditions does the population density $u$ converge uniformly to 1 in the long-time limit in a chemoattraction-repulsion system on the half-line?
• RQ2 How do the competing effects of chemoattraction ($\chi_1 > 0$) and chemorepulsion ($\chi_2 > 0$) influence the asymptotic stability of the population?
• RQ3 What role does the fast-diffusion assumption on the chemical signals ($v_1, v_2$) play in stabilizing the system and enabling convergence to equilibrium?
• RQ4 How does the time-dependent logistic source $a(t,x) - b(t,x)u$ affect the long-term dynamics when $a$ and $b$ are bounded and bounded away from zero?
• RQ5 Can the limiting behavior of the free boundary problem be rigorously captured by studying the corresponding system on the unbounded domain $\mathbb{R}^+$?

Key findings

• The population density $u(t,x)$ converges uniformly to 1 as $t \to \infty$, provided the net chemotactic effect satisfies $b_{\inf} + \chi_2\mu_2 - \chi_1\mu_1 > 0$.
• The convergence rate is quantitatively controlled: $\|u(t,\cdot) - 1\|_{\infty} \leq \left(\frac{K}{b_{\inf} + \chi_2\mu_2 - \chi_1\mu_1}\right)^n \cdot C_n + \varepsilon$ for any $n \geq 1$, with $K$ and $C_n$ depending on initial data and system parameters.
• The deviation of the chemical concentrations $v_1$ and $v_2$ from their equilibrium values is bounded by a term decaying geometrically in time, enabling uniform control of the nonlinear chemotaxis terms.
• The system exhibits asymptotic stabilization even under time-dependent logistic growth, due to the balance between proliferation and chemotactic drift.
• The results confirm that the half-line system serves as a valid formal limit of the free boundary problem, justifying its use in studying spreading-vanishing dichotomies.
• The analysis establishes that the equilibrium $u \equiv 1$ is globally attractive under the given conditions, despite the presence of nonlinear, non-local chemotactic fluxes.