Global existence and steady states of a two competing species Keller--Segel chemotaxis model

Qi Wang; Lu  Zhang; Jingyue  Yang; Jia  Hu

doi:10.3934/krm.2015.8.777

Article Contents

2015, Volume 8, Issue 4: 777-807. Doi: 10.3934/krm.2015.8.777

This issue Previous Article Strong solutions to compressible barotropic viscoelastic flow with vacuum Next Article

Global existence and steady states of a two competing species Keller--Segel chemotaxis model

1.
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130

Received: June 30, 2014

Revised: December 31, 2014

Published: November 2015

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Abstract

We study a one--dimensional quasilinear system proposed by J. Tello and M. Winkler [27] which models the population dynamics of two competing species attracted by the same chemical. The kinetic terms of the interacting species are chosen to be of the Lotka--Volterra type and the boundary conditions are of homogeneous Neumann type which represent an enclosed domain. We prove the global existence and boundedness of classical solutions to the fully parabolic system. Then we establish the existence of nonconstant positive steady states through bifurcation theory. The stability or instability of the bifurcating solutions is investigated rigorously. Our results indicate that small intervals support stable monotone positive steady states and large intervals support nonmonotone steady states. Finally, we perform extensive numerical studies to demonstrate and verify our theoretical results. Our numerical simulations also illustrate the formation of stable steady states and time--periodic solutions with various interesting spatial structures.

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Mathematics Subject Classification: Primary: 92C17, 35B32, 35B35, 35B36, 35J47, 35K20, 37K45, 37K50.

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