Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals

Pages: 1253 - 1272, Volume 22, Issue 4, June 2017      doi:10.3934/dcdsb.2017061

       Abstract        References        Full Text (475.3K)       Related Articles       

Tobias Black - Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany (email)

Abstract: We consider the two-species-two-chemical chemotaxis system \begin{align*} \left\{\begin{array}{r@{\,}l@{\quad}l@{\,}c} u_{t}\ &=\Delta u-\chi_1\nabla\!\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\ &x\in\Omega,& t>0,\\ v_{t}\ &=\Delta v-v+w,\ &x\in\Omega,& t>0,\\ w_{t}\ &=\Delta w-\chi_2\nabla\!\cdot(w\nabla z)+\mu_2 w(1-w-a_2u),\ &x\in\Omega,& t>0,\\ z_{t}\ &=\Delta z-z+u,\ &x\in\Omega,& t>0,\\ \end{array}\right. \end{align*} where $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary. The system models Lotka-Volterra competition of two species coupled with an additional chemotactic influence. In this model each species is attracted by the signal produced by the other.
    We firstly show that if $n=2$ and the parameters in the system above are positive, the solution to the corresponding Neumann initial-boundary value problem, emanating from appropriately regular and nonnegative initial data, is global and bounded.
    Furthermore, we prove asymptotic stabilization of arbitrary global bounded solutions for any $n\geq2$, in the sense that:
    • If $a_1<1$, $a_2<1$ and both $\frac{\mu_1}{\chi_1^2}$ and $\frac{\mu_2}{\chi_2^2}$ are sufficiently large, then any global solution satisfying $u\not\equiv0\not\equiv w$ converges towards the unique positive spatially homogeneous equilibrium of the system given above.
    • If $a_1\geq 1$, $a_2<1$ and $\frac{\mu_2}{\chi_2^2}$ is sufficiently large any global solution satisfying $w\not\equiv0$ tends to $(0,1,1,0)$ as $t\to\infty$.

Keywords:  Multi-species chemotaxis, boundedness, logistic source, Lotka-Volterra competition, stability.
Mathematics Subject Classification:  Primary: 35K35; Secondary: 35A01, 35B40, 35B35, 35Q92, 92C17, 92D40.

Received: June 2016;      Revised: June 2016;      Available Online: February 2017.