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    Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity
July  2018, 38(7): 3617-3636. doi: 10.3934/dcds.2018156

Global dynamics in a two-species chemotaxis-competition system with two signals

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

2. 

Depart. of Appl. Math., Chongqing Univ. of Posts and Telecommun., Chongqing 400065, China

3. 

College of Economic Math., Southwestern Univ. of Finance and Economics, Chengdu 611130, China

* Corresponding author: Xinyu Tu

Received  November 2017 Revised  February 2018 Published  April 2018

Fund Project: The first author is partially supported by the China Scholarship Council (201706050065). The second author is partially supported by National Natural Science Foundation of China (Grant Nos: 11771062, 11371384, 11571062), the Basic and Advanced Research Project of CQCSTC (Grant No: cstc2015jcyjBX0007). Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042). The fourth author is partially supported by Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006).

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type
$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$
with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain
$Ω\subset R^n$
,
$n≥2$
, where
$χ_{i}$
,
$μ_{i}$
and
$a_{i}$
$(i = 1, 2)$
are positive constants. It is shown that for any positive parameters
$χ_{i}$
,
$μ_{i}$
,
$a_{i}$
$(i = 1, 2)$
and any suitably regular initial data
$(u_{0}, w_{0})$
, this system possesses a global bounded classical solution provided that
$\frac{χ_{i}}{μ_{i}}$
are small. Moreover, when
$a_{1}, a_{2}∈ (0, 1)$
and the parameters
$μ_{1}$
and
$μ_{2}$
are sufficiently large, it is proved that the global solution
$(u, v, w, z)$
of this system exponentially approaches to the steady state
$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$
in the norm of
$L^{∞}(Ω)$
as
$t\to ∞$
; If
$a_{1}≥1>a_{2}>0$
and
$μ_{2}$
is sufficiently large, the solution of the system converges to the constant stationary solution
$\left(0, 1, 1, 0\right)$
as time tends to infinity, and the convergence rates can be calculated accurately.
Citation: Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156
References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

T. BlackJ. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.

[3]

J. CaoW. Wang and H. Yu, Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392.  doi: 10.1016/j.jmaa.2015.11.058.

[4]

E. E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.  doi: 10.1016/j.aml.2014.04.007.

[5]

A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969.

[6]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.

[7]

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

[8]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.

[9]

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.

[10]

D. Horstmann, Generaizing 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.

[11]

M. W. Htwe and Y.F Wang, Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83.  doi: 10.1016/j.crma.2016.10.024.

[12]

J. HuQ. WangJ. Yang and L. Zhang, Globale 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.

[13]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.

[14]

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

[15]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[16]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.

[17]

Y. Li, Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304.  doi: 10.1016/j.jmaa.2015.04.052.

[18]

K. Lin and C. L. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.

[19]

K. LinC. L. Mu and L. C. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.

[20]

K. LinC. L. Mu and L. C. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.

[21]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.

[22]

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.

[23]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.

[24]

M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164. 

[25]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.

[26]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.  doi: 10.1016/j.jde.2016.05.008.

[27]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[28]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.

[29]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[30]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.

[31]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[32]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469. 

[33]

K. OsakiT. TsujikawaT. A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[34]

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.

[35]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[36]

Y. S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.

[37]

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.

[38]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[39]

Q. WangJ. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.  doi: 10.3934/dcdsb.2017179.

[40]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[41]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[42]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.

[43]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[44]

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.

[45]

M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[46]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

[47]

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.

[48]

J. Zheng, Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480. 

[49]

P. Zheng and C. L. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.  doi: 10.1007/s10440-016-0083-0.

[50]

P. Zheng, C. L. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals J. Math. Phys. 58 (2017), 111501, 17pp. doi: 10.1063/1.5010681.

[51]

P. ZhengC. L. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558. 

show all references

References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

T. BlackJ. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.

[3]

J. CaoW. Wang and H. Yu, Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392.  doi: 10.1016/j.jmaa.2015.11.058.

[4]

E. E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.  doi: 10.1016/j.aml.2014.04.007.

[5]

A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969.

[6]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.

[7]

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

[8]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.

[9]

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.

[10]

D. Horstmann, Generaizing 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.

[11]

M. W. Htwe and Y.F Wang, Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83.  doi: 10.1016/j.crma.2016.10.024.

[12]

J. HuQ. WangJ. Yang and L. Zhang, Globale 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.

[13]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.

[14]

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

[15]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[16]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.

[17]

Y. Li, Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304.  doi: 10.1016/j.jmaa.2015.04.052.

[18]

K. Lin and C. L. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.

[19]

K. LinC. L. Mu and L. C. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.

[20]

K. LinC. L. Mu and L. C. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.

[21]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.

[22]

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.

[23]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.

[24]

M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164. 

[25]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.

[26]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.  doi: 10.1016/j.jde.2016.05.008.

[27]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[28]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.

[29]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[30]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.

[31]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[32]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469. 

[33]

K. OsakiT. TsujikawaT. A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[34]

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.

[35]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[36]

Y. S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.

[37]

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.

[38]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[39]

Q. WangJ. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.  doi: 10.3934/dcdsb.2017179.

[40]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[41]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[42]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.

[43]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[44]

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.

[45]

M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[46]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

[47]

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.

[48]

J. Zheng, Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480. 

[49]

P. Zheng and C. L. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.  doi: 10.1007/s10440-016-0083-0.

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