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:
[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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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).

[20]

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

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[44]

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

[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.

[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.

[47]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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).

[20]

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

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[44]

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

[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.

[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.

[47]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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
[1]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[2]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[3]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[4]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268

[5]

Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056

[6]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[7]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097

[8]

Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 269-278. doi: 10.3934/dcdss.2020015

[9]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220

[10]

Daniel Wetzel. Pattern analysis in a benthic bacteria-nutrient system. Mathematical Biosciences & Engineering, 2016, 13 (2) : 303-332. doi: 10.3934/mbe.2015004

[11]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 941-973. doi: 10.3934/dcds.2011.31.941

[12]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657

[13]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[14]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[15]

Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a two-species kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359-380. doi: 10.3934/krm.2015.8.359

[16]

Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907

[17]

Chufen Wu, Dongmei Xiao, Xiao-Qiang Zhao. Asymptotic pattern of a migratory and nonmonotone population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1171-1195. doi: 10.3934/dcdsb.2014.19.1171

[18]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

[19]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

[20]

Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (33)
  • HTML views (26)
  • Cited by (1)

Other articles
by authors

[Back to Top]