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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

Qi Wang 1, , Lu Zhang 1, , Jingyue Yang 1, and  Jia Hu 1,

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.
Keywords: two species chemotaxis model., Global existence, stationary solutions.
Mathematics Subject Classification: Primary: 92C17, 35B32, 35B35, 35B36, 35J47, 35K20, 37K45, 37K5.
Citation: Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & 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.  doi: 10.1126/science.184.4143.1292.  Google Scholar

[2]

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

[3]

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

[4]

________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, Function Spaces, 133 (1993), 9.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[5]

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

[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.  doi: 10.1017/S0956792511000258.  Google Scholar

[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.  doi: 10.1017/S0956792512000411.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[14]

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

[15]

T. Kato, Functional Analysis,, Springer Classics in Mathematics, (1995).   Google Scholar

[16]

F. Kelly, K. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Microbial Ecology, 16 (1988), 115.  doi: 10.1007/BF02018908.  Google Scholar

[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.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[19]

D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics,, Microbial Ecology, 22 (1991), 175.  doi: 10.1007/BF02540222.  Google Scholar

[20]

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

[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.  doi: 10.1002/bit.260250902.  Google Scholar

[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.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[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.  doi: 10.1137/110843964.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2015.35.1239.  Google Scholar

[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, ().   Google Scholar

[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.   Google Scholar

[33]

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

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.  doi: 10.1126/science.184.4143.1292.  Google Scholar

[2]

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

[3]

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

[4]

________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, Function Spaces, 133 (1993), 9.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[5]

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

[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.  doi: 10.1017/S0956792511000258.  Google Scholar

[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.  doi: 10.1017/S0956792512000411.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[14]

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

[15]

T. Kato, Functional Analysis,, Springer Classics in Mathematics, (1995).   Google Scholar

[16]

F. Kelly, K. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Microbial Ecology, 16 (1988), 115.  doi: 10.1007/BF02018908.  Google Scholar

[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.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[19]

D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics,, Microbial Ecology, 22 (1991), 175.  doi: 10.1007/BF02540222.  Google Scholar

[20]

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

[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.  doi: 10.1002/bit.260250902.  Google Scholar

[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.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[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.  doi: 10.1137/110843964.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2015.35.1239.  Google Scholar

[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, ().   Google Scholar

[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.   Google Scholar

[33]

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

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