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June  2017, 22(4): 1253-1272. doi: 10.3934/dcdsb.2017061

Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals

Tobias Black

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Received  June 2016 Revised  June 2016 Published  February 2017

We consider the two-species-two-chemical chemotaxis system
$\left\{ \begin{array}{l}{u_t}\; = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u(1 - u - {a_1}w),\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{u_t}\; = \Delta v - v + w,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{w_t}\; = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w(1 - w - {a_2}u),\;\;\;x \in \Omega ,t > 0,\\{z_t}\; = \Delta z - z + u,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\end{array} \right.$
where
$Ω\subset\mathbb{R}^n$
is a bounded domain with smooth boundary. The system models Lotka-Volterra competition of two species coupled with an additional chemotactic influence. In this model each species is attracted by the signal produced by the other.
We firstly show that if
$n=2$
and the parameters in the system above are positive, the solution to the corresponding Neumann initial-boundary value problem, emanating from appropriately regular and nonnegative initial data, is global and bounded.
Furthermore, we prove asymptotic stabilization of arbitrary global bounded solutions for any
$n≥q2$
, in the sense that:
• If
$a_1<1$
,
$a_2<1$
and both
$\frac{μ_1}{χ_1^2}$
and
$\frac{μ_2}{χ_2^2}$
are sufficiently large, then any global solution satisfying
$u\not\equiv0\not\equiv w$
converges towards the unique positive spatially homogeneous equilibrium of the system given above.
and
• If
$a_1≥q 1$
,
$a_2<1$
and
$\frac{μ_2}{χ_2^2}$
is sufficiently large any global solution satisfying
$w\not\equiv0$
tends to
$(0,1,1,0)$
as
$t\to∞$
.
Keywords: Multi-species chemotaxis, boundedness, logistic source, Lotka-Volterra competition, stability.
Mathematics Subject Classification: Primary:35K35;Secondary:35A01, 35B40, 35B35, 35Q92, 92C17, 92D40.
Citation: Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
References:
[1]

N. Bellomo, A. Bellouquid, Y. 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

[2]

P. Biler, E.E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Comm. Pure Appl. Math., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89. Google Scholar

[3]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193. doi: 10.1016/j.na.2012.04.035. Google Scholar

[4]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5. Google Scholar

[5]

T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860–876, arXiv: 1604.03529. doi: 10.1093/imamat/hxw036. Google Scholar

[6]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities Calc. Var. Partial Differential Equations, 55 (2016), 39 pp, arXiv: 1601.03897. doi: 10.1007/s00526-016-1027-2. Google Scholar

[7]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013. Google Scholar

[8]

J. H. Connell, The influence of interspecific competition and other factors on the distribution of the barnacle chthamalus stellatus, Ecology, 42 (1961), 710-723. doi: 10.2307/1933500. Google Scholar

[9]

D. D. Haroske and H. Triebel, Distributions, Sobolev spaces, Elliptic Equations EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. doi: 10.4171/042. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. Google Scholar

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Translations of mathematical monographs, American Mathematical Society, 1968. Google Scholar

[15]

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

[16]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. 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]

G. M. Lieberman, Second Order Parabolic Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar

[19]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[20]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, Preprint.Google Scholar

[21]

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

[22]

J. D. Murray, Mathematical Biology. I Springer-Verlag, New York, 2002. doi: 10.1007/b98868. Google Scholar

[23]

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

[24]

E. Nakaguchi and M. Efendiev, On a new dimension estimate of the global attractor for chemotaxis-growth systems, Osaka J. Math., 45 (2008), 273-281. Google Scholar

[25]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in $L^q$ for bounded and exterior domains. Applications, in Nonlinear Analysis, Function Spaces and Applications Vol. 4 (eds. M. Krbec, A. Kufner, B. Opic and J. Rákosník), Teubner-Texte Math., Vieweg+Teubner Verlag, 119 (1990), 180-223. doi: 10.1007/978-3-663-01272-6_7. Google Scholar

[30]

C. Stinner, J. 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

[31]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[32]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014. Google Scholar

[33]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys. , 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. 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. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.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, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar

[39]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar

[40]

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

[41]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. Google Scholar

[42]

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

[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, 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

[45]

M. Winkler and X. Bai, 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

[46]

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. Bellomo, A. Bellouquid, Y. 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

[2]

P. Biler, E.E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Comm. Pure Appl. Math., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89. Google Scholar

[3]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193. doi: 10.1016/j.na.2012.04.035. Google Scholar

[4]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5. Google Scholar

[5]

T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860–876, arXiv: 1604.03529. doi: 10.1093/imamat/hxw036. Google Scholar

[6]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities Calc. Var. Partial Differential Equations, 55 (2016), 39 pp, arXiv: 1601.03897. doi: 10.1007/s00526-016-1027-2. Google Scholar

[7]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013. Google Scholar

[8]

J. H. Connell, The influence of interspecific competition and other factors on the distribution of the barnacle chthamalus stellatus, Ecology, 42 (1961), 710-723. doi: 10.2307/1933500. Google Scholar

[9]

D. D. Haroske and H. Triebel, Distributions, Sobolev spaces, Elliptic Equations EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. doi: 10.4171/042. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. Google Scholar

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Translations of mathematical monographs, American Mathematical Society, 1968. Google Scholar

[15]

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

[16]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. 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]

G. M. Lieberman, Second Order Parabolic Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar

[19]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[20]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, Preprint.Google Scholar

[21]

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

[22]

J. D. Murray, Mathematical Biology. I Springer-Verlag, New York, 2002. doi: 10.1007/b98868. Google Scholar

[23]

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

[24]

E. Nakaguchi and M. Efendiev, On a new dimension estimate of the global attractor for chemotaxis-growth systems, Osaka J. Math., 45 (2008), 273-281. Google Scholar

[25]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in $L^q$ for bounded and exterior domains. Applications, in Nonlinear Analysis, Function Spaces and Applications Vol. 4 (eds. M. Krbec, A. Kufner, B. Opic and J. Rákosník), Teubner-Texte Math., Vieweg+Teubner Verlag, 119 (1990), 180-223. doi: 10.1007/978-3-663-01272-6_7. Google Scholar

[30]

C. Stinner, J. 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

[31]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[32]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014. Google Scholar

[33]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys. , 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. 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. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.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, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar

[39]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar

[40]

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

[41]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. Google Scholar

[42]

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

[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, 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

[45]

M. Winkler and X. Bai, 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

[46]

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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