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Stability analysis of an enteropathogen population growing within a heterogeneous group of animals
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 |
$\left\{ \begin{array}{l}{u_t}\; = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u(1 - u - {a_1}w),\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{u_t}\; = \Delta v - v + w,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{w_t}\; = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w(1 - w - {a_2}u),\;\;\;x \in \Omega ,t > 0,\\{z_t}\; = \Delta z - z + u,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\end{array} \right.$ |
$Ω\subset\mathbb{R}^n$ |
$n=2$ |
$n≥q2$ |
$a_1<1$ |
$a_2<1$ |
$\frac{μ_1}{χ_1^2}$ |
$\frac{μ_2}{χ_2^2}$ |
$u\not\equiv0\not\equiv w$ |
$a_1≥q 1$ |
$a_2<1$ |
$\frac{μ_2}{χ_2^2}$ |
$w\not\equiv0$ |
$(0,1,1,0)$ |
$t\to∞$ |
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. |
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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