`a`
Kinetic and Related Models (KRM)
 

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

Pages: 777 - 807, Volume 8, Issue 4, December 2015      doi:10.3934/krm.2015.8.777

 
       Abstract        References        Full Text (884.7K)       Related Articles       

Qi Wang - Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China (email)
Lu Zhang - Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China (email)
Jingyue Yang - Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China (email)
Jia Hu - Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China (email)

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.

Keywords:  Global existence, stationary solutions, two species chemotaxis model.
Mathematics Subject Classification:  Primary: 92C17, 35B32, 35B35, 35B36, 35J47, 35K20, 37K45, 37K50.

Received: July 2014;      Revised: January 2015;      Available Online: July 2015.

 References