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

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
$t\to∞$
.
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. 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.

[2]

P. BilerE.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.

[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.

[4]

P. BilerW. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

K. KutoK. OsakiT. 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.

[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.

[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.

[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.

[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]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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.

[31]

C. StinnerC. 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.

[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.

[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.

[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.

[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. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.

[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, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[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.

[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.

[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.

[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.

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

[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.

[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.

show all references

References:
[1]

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.

[2]

P. BilerE.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.

[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.

[4]

P. BilerW. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

K. KutoK. OsakiT. 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.

[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.

[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.

[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.

[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]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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.

[31]

C. StinnerC. 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.

[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.

[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.

[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.

[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. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.

[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, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[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.

[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.

[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.

[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.

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

[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.

[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.

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