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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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

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