2015, 8(5): 1023-1034. doi: 10.3934/dcdss.2015.8.1023

Stationary solutions for some shadow system of the Keller-Segel model with logistic growth

1. 

Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192, Japan, Japan

2. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, Tokyo, 182-8585, Japan

3. 

Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan

Received  December 2013 Revised  March 2015 Published  July 2015

From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.
Citation: Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023
References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015.

[2]

W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691. doi: 10.1007/BF00275511.

[3]

P. Biler, Local and global solvability of some parabolic system modelling chemotaxis,, Adv. Mathe. Sci. and Appl., 8 (1998), 715.

[4]

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type,, Appl. Anal., 4 (1974), 17. doi: 10.1080/00036817408839081.

[5]

E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO 2000,, Continuation and bifurcation software for ordinary differential equations., ().

[6]

S.-I. Ei, H. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth,, Physica D, 277 (2014), 1. doi: 10.1016/j.physd.2014.03.002.

[7]

C. Gai, Q. Wang and J. Yan, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth,, preprint, ().

[8]

D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model,, Sci. Math. Jpn, 70 (2009), 205.

[9]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity,, RIMS Kokyuroku, 1025 (1998), 44.

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[11]

N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension,, Math. Sci. Appl., 29 (2008), 265.

[12]

K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern in a chemotaxis-diffusion-growth model,, Physica D, 241 (2012), 1629. doi: 10.1016/j.physd.2012.06.009.

[13]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system,, Nonlinearity, 26 (2013), 1313. doi: 10.1088/0951-7715/26/5/1313.

[14]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint,, Discrete Continuous Dynam. Systems, (2013), 455.

[15]

K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection,, J. Differential Equations, 258 (2015), 1801. doi: 10.1016/j.jde.2014.11.016.

[16]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for generalized Allen-Cahn equations with nonlocal constraint,, preprint., ().

[17]

D. A. Lauffenburger and C. R. Kennedy, Localized bacterial infection in a chemotaxis-diffusion-growth model,, J. Math. Biol., 16 (1983), 141.

[18]

C.-S. Lin, W.-N. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.

[19]

P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern formation,, Bull. Math. Biol., 53 (1991), 701.

[20]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499. doi: 10.1016/0378-4371(96)00051-9.

[21]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X.

[22]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011.

[23]

R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters,, J. Reine Angew. Math., 364 (1984), 1. doi: 10.1515/crll.1984.346.1.

[24]

R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems,, Lecture Notes in Mathematics, (1458).

[25]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis,, Adv. Math. Sci. Appl., 10 (2000), 191.

[26]

J. Shi, Semilinear Neumann boundary value problems on a rectangle,, Trans. Amer. Math. Soc., 354 (2002), 3117. doi: 10.1090/S0002-9947-02-03007-6.

[27]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003.

[28]

T. Tsujikawa, Global structure of the stationary solutions for the limiting system of a chemotaxis-growth model, to appear in, RIMS Kokyuroku, (2014).

show all references

References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015.

[2]

W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691. doi: 10.1007/BF00275511.

[3]

P. Biler, Local and global solvability of some parabolic system modelling chemotaxis,, Adv. Mathe. Sci. and Appl., 8 (1998), 715.

[4]

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type,, Appl. Anal., 4 (1974), 17. doi: 10.1080/00036817408839081.

[5]

E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO 2000,, Continuation and bifurcation software for ordinary differential equations., ().

[6]

S.-I. Ei, H. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth,, Physica D, 277 (2014), 1. doi: 10.1016/j.physd.2014.03.002.

[7]

C. Gai, Q. Wang and J. Yan, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth,, preprint, ().

[8]

D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model,, Sci. Math. Jpn, 70 (2009), 205.

[9]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity,, RIMS Kokyuroku, 1025 (1998), 44.

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[11]

N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension,, Math. Sci. Appl., 29 (2008), 265.

[12]

K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern in a chemotaxis-diffusion-growth model,, Physica D, 241 (2012), 1629. doi: 10.1016/j.physd.2012.06.009.

[13]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system,, Nonlinearity, 26 (2013), 1313. doi: 10.1088/0951-7715/26/5/1313.

[14]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint,, Discrete Continuous Dynam. Systems, (2013), 455.

[15]

K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection,, J. Differential Equations, 258 (2015), 1801. doi: 10.1016/j.jde.2014.11.016.

[16]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for generalized Allen-Cahn equations with nonlocal constraint,, preprint., ().

[17]

D. A. Lauffenburger and C. R. Kennedy, Localized bacterial infection in a chemotaxis-diffusion-growth model,, J. Math. Biol., 16 (1983), 141.

[18]

C.-S. Lin, W.-N. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.

[19]

P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern formation,, Bull. Math. Biol., 53 (1991), 701.

[20]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499. doi: 10.1016/0378-4371(96)00051-9.

[21]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X.

[22]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011.

[23]

R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters,, J. Reine Angew. Math., 364 (1984), 1. doi: 10.1515/crll.1984.346.1.

[24]

R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems,, Lecture Notes in Mathematics, (1458).

[25]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis,, Adv. Math. Sci. Appl., 10 (2000), 191.

[26]

J. Shi, Semilinear Neumann boundary value problems on a rectangle,, Trans. Amer. Math. Soc., 354 (2002), 3117. doi: 10.1090/S0002-9947-02-03007-6.

[27]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003.

[28]

T. Tsujikawa, Global structure of the stationary solutions for the limiting system of a chemotaxis-growth model, to appear in, RIMS Kokyuroku, (2014).

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