November  2017, 22(9): 3547-3574. doi: 10.3934/dcdsb.2017179

Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth

1. 

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

2. 

Department of Mathematics, Southern Methodist University, 6425 Boaz Lane, Dallas TX 75205, USA

* Corresponding author

All authors thank the two anonymous referees for their helpful suggestions.

Received  August 2016 Revised  May 2017 Published  July 2017

Fund Project: QW is partially supported by NSF-China (Grant 11501460)

This paper investigates the formation of time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time-periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the 3×3 system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.

Citation: Qi Wang, Jingyue Yang, Lu Zhang. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179
References:
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N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar

[2]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63. doi: 10.1007/BFb0083479. Google Scholar

[3]

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

[4]

H. Amann, 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. Google Scholar

[5]

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1953. xiii+166 pp. Google Scholar

[6]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

[7]

P. BilerI. Espejo and E. Guerra, Blow-up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89. Google Scholar

[8]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. Ⅰ., Colloq. Math., 66 (1994), 319-334. Google Scholar

[9]

S. Y. A. Chang and P. Yang, Conformal deformation of metric on $S^2$, J. Differential Geom., 27 (1988), 259-296. doi: 10.4310/jdg/1214441783. Google Scholar

[10]

A. ChertockA. KurganovX. 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. Google Scholar

[11]

S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations, 26 (1977), 112-159. doi: 10.1016/0022-0396(77)90101-2. Google Scholar

[12]

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

[13]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827. Google Scholar

[14]

E. N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, Methods Appl. Anal., 8 (2001), 245-256. doi: 10.4310/MAA.2001.v8.n2.a3. Google Scholar

[15]

S. I. EiH. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Phys. D, 277 (2014), 1-21. doi: 10.1016/j.physd.2014.03.002. Google Scholar

[16]

E. EspejoK. Vilches and C. Conca, 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. Google Scholar

[17]

G. Gerisch, Chemotaxis in dictyostelium, Annu. Rev. Physiol., 44 (1982), 535-552. doi: 10.1146/annurev.ph.44.030182.002535. Google Scholar

[18]

P. Haastert and P. Devreotes, Chemotaxis: Signalling the way forward, Nat. Rev. Mol. Cell Biol., 5 (2004), 626-634. doi: 10.1038/nrm1435. Google Scholar

[19]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981. v+311 pp. (microfiche insert). Google Scholar

[20]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. Google Scholar

[21]

K. Hepp and E. H. Lieb, Phase transition in reservoir driven open systems with applications to lasers and superconductors, Condensed Matter Physics and Exactly Soluble Models, (2004), 145--175. doi: 10.1007/978-3-662-06390-3_13. Google Scholar

[22]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[23]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ., Jahresber DMV, 105 (2003), 103-165. Google Scholar

[24]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ., Jahresber DMV, 106 (2004), 51-69. Google Scholar

[25]

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. Google Scholar

[26]

G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problémes d'evolution du type Navier-Stokes, Arch. Rational Mech. Anal., 47 (1972), 301-329. doi: 10.1007/BF00281637. Google Scholar

[27]

V. Iudovic, Stability of steady flows of viscous incompressible fluids, Soviet Physics Dokl., 10 (1965), 293-295. Google Scholar

[28]

V. Iudovic, On the stability of self-oscillations of a liquid, Soviet Physics Dokl., 11 (1970), 1543-1546. Google Scholar

[29]

V. Iudovic, Appearance of auto-oscillations in a fluid, Prikl. Mat. Meh., 35 (1971), 638-655. doi: 10.1016/0021-8928(71)90053-0. Google Scholar

[30]

D. D. Joseph, Stability of Fluid Motions. I. , Springer Tracts in Natural Philosophy, Vol. 27. Springer-Verlag, Berlin-New York, 1976. xiii+282 pp. doi: 10.1007/978-3-642-80991-0. Google Scholar

[31]

D. D. Joseph and D. Nield, Stability of bifurcating time-periodic and steady solutions of arbitrary amplitude, Arch. Rational Mech. Anal., 58 (1975), 369-380. doi: 10.1007/BF00250296. Google Scholar

[32]

D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 79-109. doi: 10.1007/BF00253039. Google Scholar

[33]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp. ISBN: 3-540-58661-X doi: 10.1007/978-3-642-66282-9. Google Scholar

[34]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[35]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[36]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, J. Theoret. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[37]

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. Google Scholar

[38]

O. A. Ladyenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, American Mathematical Society, (1967), 736pp. Google Scholar

[39]

P. LiuJ. 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. Google Scholar

[40]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. Google Scholar

[41]

J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Lecture Notes in Appl. Math. Sci. , 18 Springer-Verlag, Berlin and New York, 1976. Google Scholar

[42]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar

[43]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. Google Scholar

[44]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[45]

D. H. Sather, Bifurcation of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 41 (1971), 68-80. doi: 10.1007/BF00250178. Google Scholar

[46]

D. H. Sattinger, Bifurcation and symmetry breaking in applied mathematics, Bull. Amer. Math. Soc., 3 (1980), 779-819. doi: 10.1090/S0273-0979-1980-14823-5. Google Scholar

[47]

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

[48]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar

[49]

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. Google Scholar

[50]

Z. A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. Google Scholar

[51]

Q. WangC. 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. Google Scholar

[52]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777. Google Scholar

[53]

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. Google Scholar

[54]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084. Google Scholar

show all references

References:
[1]

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

[2]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63. doi: 10.1007/BFb0083479. Google Scholar

[3]

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

[4]

H. Amann, 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. Google Scholar

[5]

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1953. xiii+166 pp. Google Scholar

[6]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

[7]

P. BilerI. Espejo and E. Guerra, Blow-up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89. Google Scholar

[8]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. Ⅰ., Colloq. Math., 66 (1994), 319-334. Google Scholar

[9]

S. Y. A. Chang and P. Yang, Conformal deformation of metric on $S^2$, J. Differential Geom., 27 (1988), 259-296. doi: 10.4310/jdg/1214441783. Google Scholar

[10]

A. ChertockA. KurganovX. 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. Google Scholar

[11]

S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations, 26 (1977), 112-159. doi: 10.1016/0022-0396(77)90101-2. Google Scholar

[12]

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

[13]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827. Google Scholar

[14]

E. N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, Methods Appl. Anal., 8 (2001), 245-256. doi: 10.4310/MAA.2001.v8.n2.a3. Google Scholar

[15]

S. I. EiH. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Phys. D, 277 (2014), 1-21. doi: 10.1016/j.physd.2014.03.002. Google Scholar

[16]

E. EspejoK. Vilches and C. Conca, 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. Google Scholar

[17]

G. Gerisch, Chemotaxis in dictyostelium, Annu. Rev. Physiol., 44 (1982), 535-552. doi: 10.1146/annurev.ph.44.030182.002535. Google Scholar

[18]

P. Haastert and P. Devreotes, Chemotaxis: Signalling the way forward, Nat. Rev. Mol. Cell Biol., 5 (2004), 626-634. doi: 10.1038/nrm1435. Google Scholar

[19]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981. v+311 pp. (microfiche insert). Google Scholar

[20]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. Google Scholar

[21]

K. Hepp and E. H. Lieb, Phase transition in reservoir driven open systems with applications to lasers and superconductors, Condensed Matter Physics and Exactly Soluble Models, (2004), 145--175. doi: 10.1007/978-3-662-06390-3_13. Google Scholar

[22]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[23]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ., Jahresber DMV, 105 (2003), 103-165. Google Scholar

[24]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ., Jahresber DMV, 106 (2004), 51-69. Google Scholar

[25]

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. Google Scholar

[26]

G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problémes d'evolution du type Navier-Stokes, Arch. Rational Mech. Anal., 47 (1972), 301-329. doi: 10.1007/BF00281637. Google Scholar

[27]

V. Iudovic, Stability of steady flows of viscous incompressible fluids, Soviet Physics Dokl., 10 (1965), 293-295. Google Scholar

[28]

V. Iudovic, On the stability of self-oscillations of a liquid, Soviet Physics Dokl., 11 (1970), 1543-1546. Google Scholar

[29]

V. Iudovic, Appearance of auto-oscillations in a fluid, Prikl. Mat. Meh., 35 (1971), 638-655. doi: 10.1016/0021-8928(71)90053-0. Google Scholar

[30]

D. D. Joseph, Stability of Fluid Motions. I. , Springer Tracts in Natural Philosophy, Vol. 27. Springer-Verlag, Berlin-New York, 1976. xiii+282 pp. doi: 10.1007/978-3-642-80991-0. Google Scholar

[31]

D. D. Joseph and D. Nield, Stability of bifurcating time-periodic and steady solutions of arbitrary amplitude, Arch. Rational Mech. Anal., 58 (1975), 369-380. doi: 10.1007/BF00250296. Google Scholar

[32]

D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 79-109. doi: 10.1007/BF00253039. Google Scholar

[33]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp. ISBN: 3-540-58661-X doi: 10.1007/978-3-642-66282-9. Google Scholar

[34]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[35]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[36]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, J. Theoret. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[37]

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. Google Scholar

[38]

O. A. Ladyenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, American Mathematical Society, (1967), 736pp. Google Scholar

[39]

P. LiuJ. 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. Google Scholar

[40]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. Google Scholar

[41]

J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Lecture Notes in Appl. Math. Sci. , 18 Springer-Verlag, Berlin and New York, 1976. Google Scholar

[42]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar

[43]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. Google Scholar

[44]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[45]

D. H. Sather, Bifurcation of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 41 (1971), 68-80. doi: 10.1007/BF00250178. Google Scholar

[46]

D. H. Sattinger, Bifurcation and symmetry breaking in applied mathematics, Bull. Amer. Math. Soc., 3 (1980), 779-819. doi: 10.1090/S0273-0979-1980-14823-5. Google Scholar

[47]

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

[48]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar

[49]

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. Google Scholar

[50]

Z. A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. Google Scholar

[51]

Q. WangC. 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. Google Scholar

[52]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777. Google Scholar

[53]

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. Google Scholar

[54]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084. Google Scholar

Figure 1.  Bifurcation diagrams of $\rho_k(s)$ around $(\bar u,\bar v,\bar w)$. The stable bifurcation curve is plotted in solid lines and the unstable bifurcation curve is plotted in imaginary line. The branch $\rho_{k}(s)$ around $(\bar u,\bar v,\bar w,\chi_{k})$ is always unstable if $k\neq k_0$, while the turning direction of $\rho_{k_0}(s)$ determines its stability
Figure 2.  Initiation and development of time-periodic spatial patterns to (1.1) over $(0,6)$ with initial data being small perturbations of $(\bar u,\bar v,\bar w)$. System parameters are chosen to be $d_1=5$, $d_2=0.1$, $\mu_1=\mu_2=1$, $\lambda=5$, $\xi=0.1$ and $\chi=80$. Our theoretical results indicate that the homogeneous equilibrium loses its stability at $\chi_0=\chi^H_{2}\approx 63.2$ through Hopf bifurcation to a stable time-periodic pattern which has spatial profile $\cos \frac{\pi x}{3}$ and period $T\approx 8$. Space and time grid sizes are $\Delta x=0.02$ and $\Delta t=0.05$. The numerical simulations are in good agreement with our theoretical findings
Figure 3.  In each subfigure, we plot in the 3D $u$-$v$-$w$ phase space the trajectories for specific locations $x=1,2,...6$ which converge to enclosed orbits. $\Delta x=0.02$ and $\Delta t=0.05$
Figure 4.  Effect of cellular growth on the pattern formation of $u$-species, where we choose $\mu_1=\mu_2$. System parameters are chosen to be $d_1=8$, $d_2=0.5$, $\chi=130$ and $\xi=0.4$. Initial data are taken to be small perturbations of $(\bar u,\bar v,\bar w)$. Space and time grid sizes are $\Delta x=L/500=0.012$ and $\Delta t=0.05$. We observe that the cellular growth rate $\mu$ supports the formation of periodic patterns. However, the periodic pattern disappears at $\mu\approx 2.1$, for which we surmise that the oscillating solutions become unstable and develop into a stable stationary pattern
Figure 5.  Effect of domain size on the pattern formation of $u$-species. We choose the system parameters to be the same as those in Figure 3 except that $\chi$ is slightly larger than $\chi_{k_0}$. $\Delta x=L/500$ and $\Delta t=0.05$ in each graph. Our simulations support our theoretical findings that large domains support periodic patterns with higher modes, however when the domain size is small, therefore does not exist time-periodic solutions that bifurcate from the homogeneous solution
Figure 6.  Pattern formation of $u$-species in (2.1) when chemotaxis rate $\chi$ is far away from $\chi_{k_0}$=63.2
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