Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 462, Issue 1, Pages 1048-1072Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2018.01.023
Keywords
Instability; Spectral analysis; Cross-diffusion system; Asymptotic behavior
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Funding
- National Natural Science Foundation of China [11471221, 11501031]
- Beijing Municipal Education Commission [KZ201310028030]
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This paper is concerned with the existence and the stability of steady state solutions for the SKT biological competition model with cross-diffusion. By applying the higher order expansion and some detailed spectral analysis to the limiting system as the cross diffusion rate tends to infinite, it is proved that the nontrivial positive steady states with some special bifurcating structure are unstable. Further, the existence and the instability of the corresponding nontrivial positive steady states for the original cross-diffusion system are proved by applying perturbation argument. Finally, we show the global existence of solutions for the limiting system and present some numerical simulation on the large time behavior of the solution with more general initial data. (C) 2018 Elsevier Inc. All rights reserved.
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