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

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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

$(0,1,1,0)$

$t\to∞$

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:

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[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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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