Journal
KINETIC AND RELATED MODELS
Volume 8, Issue 4, Pages 777-807Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/krm.2015.8.777
Keywords
Global existence; stationary solutions; two species chemotaxis model
Categories
Funding
- SRF for ROCS, SEM
- Department of Education, Sichuan China [15ZA0382]
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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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