Global existence and steady states of a two competing species KellerSegel chemotaxis model
Pages: 777  807,
Issue 4,
December
2015
doi:10.3934/krm.2015.8.777 Abstract
References
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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)
1 
J. Adler and W. Tso, Decision making in bacteria: Chemotactic response of Escherichia coli to conflic stimuli, Science, 184 (1974), 12921294. 

2 
N. D. Alikakos, $L^p$ bounds of solutions of reactiondiffusion equations, Comm. Partial Differential Equations, 4 (1979), 827868. 

3 
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reactiondiffusion systems, Differential Integral Equations, 3 (1990), 1375. 

4 
________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9126. 

5 
P. Biler, E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal, 12 (2013), 8998. 

6 
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic KellerSegel system in $\mathbbR^2$, European J. Appl. Math, 22 (2011), 553580. 

7 
_______, Sharp Condition for blowup and global existence in a two species chemotactic KellerSegel system in $\mathbb R^ 2$, European J. Appl. Math, 24 (2013), 297313. 

8 
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 5195. 

9 
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321340. 

10 
________, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161180. 

11 
E. Espejo, A. Stevens and J. J. L. VelÃ¡zquez, Simultaneous finite time blowup in a twospecies model for chemotaxis, Analysis, 29 (2009), 317338. 

12 
D. Henry, Geometric Theory of Semilinear Parabolic Equations, SpringerVerlag, BerlinNew York, 1981. 

13 
D. Horstmann, Generalizing the KellerSegel model: Lyapunov functionals, steady state analysis, and blowup results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci, 21 (2011), 231270. 

14 
D. Horstmann and M. Winkler, Boundedness vs. blowup in a chemotaxis system, J. Differential Equations, 215 (2005), 52107. 

15 
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. 

16 
F. Kelly, K. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Microbial Ecology, 16 (1988), 115131. 

17 
K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reactiondiffusion systems in convex domains, J. Differential Equations, 58 (1985), 1521. 

18 
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and QuasiLinear Equations of Parabolic Type, American Mathematical Society, 1968, 648 pages. 

19 
D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial Ecology, 22 (1991), 175185. 

20 
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), 209219. 

21 
D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Biotechnol Bioeng., 25 (1983), 21032125. 

22 
P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attractionrepulsion KellerSegel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 25972625. 

23 
M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math, 72 (2012), 740766. 

24 
P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487513. 

25 
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 27882812. 

26 
G. Simonett, Center manifolds for quasilinear reactiondiffusion systems, Differential Integral Equations, 8 (1995), 753796. 

27 
J. I. Tello and M. Winkler, Stabilization in a twospecies chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 14131425. 

28 
N. Tsang, R. Macnab and J. Koshland, Common mechanism for repellents and attractants in bacterial chemotaxis, Science, 181 (1973), 6063. 

29 
F. Verhagen and H. Laanbroek, Competition for ammonium between nitrifying and heterotrophic bacteria in dual energylimited chemostats, Appl. and Enviro. Microbiology, 57 (1991), 32553263. 

30 
Q. Wang, C. Gai and J. Yan, Qualitative analysis of a LotkaVolterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 12391284. 

31 
Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a twocompetingspecies KellerSegel chemotaxis model: effect of cellular growth, preprint, arXiv:1505.06463 

32 
X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math, 60 (2002), 505531. 

33 
M. Winkler, Aggregation vs. global diffusive behavior in the higherdimensional KellerSegel model, J. Differential Equations, 248 (2010), 28892905. 

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