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Multi-spots Steady States in Two-species Keller-Segel Models with Logistic Growth: Large Chemotactic Attraction Regime

Fanze Kong, Juncheng Wei, Liangshun Xu

TL;DR

This work addresses pattern formation in a two-species Keller–Segel system with logistic growth in a bounded 2D domain by analyzing the singular limit of large chemotactic attraction. The authors develop an inner–outer gluing framework, leveraging Liouville system solutions and Neumann Green’s functions to construct multi-spot steady states, with spot locations tied to critical points of an interaction energy $\\mathcal{J}_m$. They prove the existence of both interior and boundary spots under a positive definite interaction matrix and provide rigorous linear theories for the inner and outer problems, followed by a nonlinear fixed-point argument to complete the construction; numerical simulations corroborate the theoretical patterns and illustrate dependence on parameters. The results extend the single-species and simpler two-species analyses by handling full coupling and logistic terms, offering insights into how inter-species interactions shape multi-spot configurations in chemotaxis-driven systems and suggesting directions for exploring non-positive interaction regimes and stability under varied diffusion and logistic rates.

Abstract

One of the most important findings in the study of chemotactic process is self-organized cellular aggregation, and a high volume of results are devoted to the analysis of a concentration of single species. Whereas, the multi-species case is not understood as well as the single species one. In this paper, we consider two-species chemotaxis systems with logistic source in a bounded domain $Ω\subset \mathbb R^2.$ Under the large chemo-attractive coefficients and one certain type of chemical production coefficient matrices, we employ the inner-outer gluing approach to construct multi-spots steady states, in which the profiles of cellular densities have strong connections with the entire solutions to Liouville systems and their locations are determined in terms of reduced-wave Green's functions. In particular, some numerical simulations and formal analysis are performed to support our rigorous studies.

Multi-spots Steady States in Two-species Keller-Segel Models with Logistic Growth: Large Chemotactic Attraction Regime

TL;DR

This work addresses pattern formation in a two-species Keller–Segel system with logistic growth in a bounded 2D domain by analyzing the singular limit of large chemotactic attraction. The authors develop an inner–outer gluing framework, leveraging Liouville system solutions and Neumann Green’s functions to construct multi-spot steady states, with spot locations tied to critical points of an interaction energy . They prove the existence of both interior and boundary spots under a positive definite interaction matrix and provide rigorous linear theories for the inner and outer problems, followed by a nonlinear fixed-point argument to complete the construction; numerical simulations corroborate the theoretical patterns and illustrate dependence on parameters. The results extend the single-species and simpler two-species analyses by handling full coupling and logistic terms, offering insights into how inter-species interactions shape multi-spot configurations in chemotaxis-driven systems and suggesting directions for exploring non-positive interaction regimes and stability under varied diffusion and logistic rates.

Abstract

One of the most important findings in the study of chemotactic process is self-organized cellular aggregation, and a high volume of results are devoted to the analysis of a concentration of single species. Whereas, the multi-species case is not understood as well as the single species one. In this paper, we consider two-species chemotaxis systems with logistic source in a bounded domain Under the large chemo-attractive coefficients and one certain type of chemical production coefficient matrices, we employ the inner-outer gluing approach to construct multi-spots steady states, in which the profiles of cellular densities have strong connections with the entire solutions to Liouville systems and their locations are determined in terms of reduced-wave Green's functions. In particular, some numerical simulations and formal analysis are performed to support our rigorous studies.

Paper Structure

This paper contains 12 sections, 7 theorems, 280 equations, 4 figures.

Table of Contents

  1. Introduction and Main Results
  2. Chemotaxis and Keller-Segel Models
  3. The Choice of Ansatz and Error Computations
  4. Inner Linear Theory
  5. Outer Linear Theory
  6. Inner-outer Gluing Procedures
  7. Construction of Interior Spots
  8. Construction of Boundary Spots
  9. Numerical Studies and Discussion
  10. Discussion
  11. Acknowledgments
  12. Formal Construction of Spots

Key Result

Theorem 1.1

Assume that $k$, $o$ are non-negative integers with $k+o\geq 1$ and $\lambda_j \bar{u}_j<\bar{C}_{j,\Omega}$, $j=1,2.$ Then for sufficiently large $\chi_1:=\chi:=\frac{1}{\varepsilon^2}$ with $\chi_2=\gamma\chi_1$ and given positive constant $\gamma$, there exists a solution $(u_{1,\chi},u_{2,\chi}, where $m_j = \sum_{l=1}^2b_{jl}\sigma_l$, $H$ is defined as the regular part of Neumann Green's fun

Figures (4)

  • Figure 1: Schematic Diagram of (\ref{['algebraicsystemforsigma']})
  • Figure 2: The numerical profile of a single boundary spot steady state obtained by using FLEXPDE7flex2021 to (\ref{['timedependent']}) with $\Omega=(0,2)\times (0,2),$ where the rest parameters are set as $\chi_1=\chi_2=8.5$, $\lambda_1=\lambda_2=0.5$, $\bar{u}_1=2$, $\bar{u}_2=1,$$a_{11}=2$, $a_{12}=1$, $a_{21}=2$ and $a_{22}=3.$ Here the initial data are chosen as $u_{10}=u_{20}=6e^{-10(x^2+y^2)}+0.1$ and $v_{10}=v_{20}=2e^{-10(x^2+y^2)}+0.1$. The numerical solution is captured by approximating the time-dependent system (\ref{['timedependent']}) with $t = 2000$.
  • Figure 3: The numerical profile of a single interior spot to (\ref{['ss']}) with $\Omega=(0,2)\times (0,2),$ where the rest parameters are the same except $\chi_1=\chi_2=1$ and $d_{v1}=d_{v2}=0.05$. Here we incorporate chemical self-diffusion rates $d_{v1}$ and $d_{v2}$ in the $v_1$-equation and $v_2$-equation of (\ref{['ss']}). In particular, the initial data are chosen as $u_{10}=u_{20}=6e^{-10[(x-1)^2+(y-1)^2]}+0.1$ and $v_{10}=v_{20}=2e^{-10[(x-1)^2+(y-1)^2]}+0.1$. The numerical results suggest that the single interior spot in (\ref{['ss']}) is locally stable.
  • Figure 4: The profiles of two double boundary spots to (\ref{['ss']}) with $\Omega=(0,2)\times (0,2).$ Here the rest parameters are the same as those stated in Figure \ref{['fig:singlespike']} except that $a_{12}=-1$ and $a_{21}=-2$. These numerical findings suggest that the locations of concentrated cellular densities $u_1$ and $u_2$ are different when the interaction coefficients are negative.

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 4 more