December  2015, 8(4): 777-807. doi: 10.3934/krm.2015.8.777

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, China, China, China

Received  July 2014 Revised  January 2015 Published  July 2015

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.
Citation: Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic and Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777
References:
[1]

J. Adler and W. Tso, Decision making in bacteria: Chemotactic response of Escherichia coli to conflic stimuli, Science, 184 (1974), 1292-1294. doi: 10.1126/science.184.4143.1292.

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.

[4]

________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[5]

P. Biler, E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal, 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89.

[6]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math, 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[7]

_______, Sharp Condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb R^ 2$, European J. Appl. Math, 24 (2013), 297-313. doi: 10.1017/S0956792512000411.

[8]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[10]

________, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.

[11]

E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[13]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[14]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[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), 115-131. doi: 10.1007/BF02018908.

[17]

K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear 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), 175-185. doi: 10.1007/BF02540222.

[20]

D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), 209-219. doi: 10.1016/S0006-3495(82)84476-7.

[21]

D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Biotechnol Bioeng., 25 (1983), 2103-2125. doi: 10.1002/bit.260250902.

[22]

P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[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), 740-766. doi: 10.1137/110843964.

[24]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[25]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.

[26]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796.

[27]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.

[28]

N. Tsang, R. Macnab and J. Koshland, Common mechanism for repellents and attractants in bacterial chemotaxis, Science, 181 (1973), 60-63. doi: 10.1126/science.181.4094.60.

[29]

F. Verhagen and H. Laanbroek, Competition for ammonium between nitrifying and heterotrophic bacteria in dual energy-limited chemostats, Appl. and Enviro. Microbiology, 57 (1991), 3255-3263.

[30]

Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.

[31]

Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth,, preprint, (). 

[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), 505-531.

[33]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

show all references

References:
[1]

J. Adler and W. Tso, Decision making in bacteria: Chemotactic response of Escherichia coli to conflic stimuli, Science, 184 (1974), 1292-1294. doi: 10.1126/science.184.4143.1292.

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.

[4]

________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[5]

P. Biler, E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal, 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89.

[6]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math, 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[7]

_______, Sharp Condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb R^ 2$, European J. Appl. Math, 24 (2013), 297-313. doi: 10.1017/S0956792512000411.

[8]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[10]

________, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.

[11]

E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[13]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[14]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[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), 115-131. doi: 10.1007/BF02018908.

[17]

K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear 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), 175-185. doi: 10.1007/BF02540222.

[20]

D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), 209-219. doi: 10.1016/S0006-3495(82)84476-7.

[21]

D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Biotechnol Bioeng., 25 (1983), 2103-2125. doi: 10.1002/bit.260250902.

[22]

P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[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), 740-766. doi: 10.1137/110843964.

[24]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[25]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.

[26]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796.

[27]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.

[28]

N. Tsang, R. Macnab and J. Koshland, Common mechanism for repellents and attractants in bacterial chemotaxis, Science, 181 (1973), 60-63. doi: 10.1126/science.181.4094.60.

[29]

F. Verhagen and H. Laanbroek, Competition for ammonium between nitrifying and heterotrophic bacteria in dual energy-limited chemostats, Appl. and Enviro. Microbiology, 57 (1991), 3255-3263.

[30]

Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.

[31]

Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth,, preprint, (). 

[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), 505-531.

[33]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

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