American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020092

Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model

Georges Chamoun 1, , Moustafa Ibrahim 2,, , Mazen Saad 3, and  Raafat Talhouk 4,

1. 

Lebanese University, Faculty of Science Ⅳ. Laboratory of mathematics-EDST, Hadath, Lebanon
2. 

College of Engineering and Technology, American University of the Middle East, Kuwait
3. 

École Centrale de Nantes. UMR 6629 CNRS, laboratoire de mathématiques Jean Leray, F-44321, Nantes, France
4. 

Lebanese University, Faculty of Science Ⅰ. Laboratory of mathematics-EDST, Hadath, Lebanon

* Corresponding author: Moustafa Ibrahim

Received  June 2019 Revised  November 2019 Published  April 2020

Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.

Keywords: Degenerate parabolic equation, Destabilization mechanism, Pattern formation, Growing modes, Nonlinear instability.
Mathematics Subject Classification: Primary: 35B40, 35K57, 35K55, 35K65, 65M08; Secondary: 70K50, 92C15.
Citation: Georges Chamoun, Moustafa Ibrahim, Mazen Saad, Raafat Talhouk. Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020092
References:
[1]

B. Andreianov, M. Bendahmane and M. Saad, Finite volume methods for degenerate chemotaxis model, J. Comput. Appl. Math., 235 (2011), 4015-4031.  doi: 10.1016/j.cam.2011.02.023.  Google Scholar

[2]

C. Bardos, Y. Guo and W. Strauss, Stable and unstable ideal plane flows, Chinese Ann. Math. Ser. B, 23 (2002), 149-164.  doi: 10.1142/S0252959902000158.  Google Scholar

[3]

C. Cancès, M. Ibrahim and M. Saad, A Nonlinear CVFE Scheme for an anisotropic degenerate nonlinear Keller-Segel model, European Consortium for Mathematics in Industry, 22 (2014), 1037-1046.  doi: 10.1007/978-3-319-23413-7_145.  Google Scholar

[4]

G. Chamoun, M. Ibrahim, M. Saad and R. Talhouk, Numerical simulation of heterogeneous steady states for a reaction-diffusion degenerate Keller-Segel model, European Consortium for Mathematics in Industry, 30 (2019), 411-417.  doi: 10.1007/978-3-030-27550-1_52.  Google Scholar

[5]

G. Chamoun, M. Saad and R. Talhouk, A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion, Comput. Math. Appl., 68 (2014), 1052-1070.  doi: 10.1016/j.camwa.2014.04.010.  Google Scholar

[6]

G. Chamoun, M. Saad and R. Talhouk, Monotone combined edge finite volume–finite element scheme for anisotropic Keller-Segel model, Numer. Methods Partial Differential Equations, 30 (2014), 1030-1065.  doi: 10.1002/num.21858.  Google Scholar

[7]

M. H. Cohen and A. Robertson, Chemotaxis and the early stages of aggregation in cellular slime molds, Journal of Theoretical Biology, 31 (1971), 119-130.  doi: 10.1016/0022-5193(71)90125-1.  Google Scholar

[8]

P. De Leenheer, J. Gopalakrishnan and E. Zuhr, Instability in a generalized Keller-Segel model, Journal of Biological Dynamics, 6 (2012), 974-991.  doi: 10.1080/17513758.2012.714478.  Google Scholar

[9]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of Numerical Analysis, 7 (2000), 713-1020.   Google Scholar

[10]

S. Fu and F. Cao, Pattern formation of a Keller-Segel model with the source term $u^p(1-u)$, J. Math., (2013), Art. ID 454513, 11 pp. doi: 10.1155/2013/454513.  Google Scholar

[11]

D. F. Griffiths and D. J. Higham, Numerical Methods For Ordinary Differential Equations. Initial Value Problems, Springer-Verlag London, Ltd., London, 2010. doi: 10.1007/978-0-85729-148-6.  Google Scholar

[12]

Y. Guo, Instability of symmetric vortices with large charge and coupling constant, Comm. Pure Appl. Math., 49 (1996), 1051-1080.  doi: 10.1002/(SICI)1097-0312(199610)49:10<1051::AID-CPA2>3.0.CO;2-D.  Google Scholar

[13]

Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354.  doi: 10.1007/s00220-003-1017-z.  Google Scholar

[14]

Y. Guo and H. J. Hwang, Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.  doi: 10.1016/j.jde.2010.07.025.  Google Scholar

[15]

Y. Guo and H. J. Hwang, Pattern formation. (Ⅱ). The Turing instability, Proc. Amer. Math. Soc., 135 (2007), 2855-2866.  doi: 10.1090/S0002-9939-07-08850-8.  Google Scholar

[16]

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

[17]

T. Hoang and H. J. Hwang, Turing instability in a general system, Nonlinear Anal., 91 (2013), 93-113.  doi: 10.1016/j.na.2013.06.010.  Google Scholar

[18]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.   Google Scholar

[19]

M. Ibrahim and M. Saad, On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model, Comput. Math. Appl., 68 (2014), 1032-1051.  doi: 10.1016/j.camwa.2014.03.010.  Google Scholar

[20]

H.-Y. Jin and Z.-A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[21]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[22]

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

[23]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. Google Scholar

[24]

P. Laurençcot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, Nonlinear Elliptic and Parabolic Problems, 64 (2005), 273-290.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar

[25]

R. J. LeVeque, Conservative methods for nonlinear problems, in Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1990,122–135. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[26]

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

[27]

P. K. Maini, The impact of Turing's work on pattern formation in biology, Mathematics Today, 40 (2004), 140-141.   Google Scholar

[28]

J. D. Murray, Mathematical biology Ⅱ: Spatial models and biomedical applications, in Interdisciplinary Applied Mathematics, vol. 18, Springer-Verlag, New York, 2003.  Google Scholar

[29]

E. Sander and T. Wanner, Pattern formation in a nonlinear model for animal coats, J. Differential Equations, 191 (2003), 143-174.  doi: 10.1016/S0022-0396(02)00156-0.  Google Scholar

[30]

Y. 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

[31]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.   Google Scholar

[32]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.  Google Scholar

[33]

S. Wu, J. Shi and B. Wu, Global existence of solutions to an attraction-repulsion chemotaxis model with growth, Commun. Pure Appl. Anal., 16 (2017), 1037-1058.  doi: 10.3934/cpaa.2017050.  Google Scholar

show all references

References:
[1]

B. Andreianov, M. Bendahmane and M. Saad, Finite volume methods for degenerate chemotaxis model, J. Comput. Appl. Math., 235 (2011), 4015-4031.  doi: 10.1016/j.cam.2011.02.023.  Google Scholar

[2]

C. Bardos, Y. Guo and W. Strauss, Stable and unstable ideal plane flows, Chinese Ann. Math. Ser. B, 23 (2002), 149-164.  doi: 10.1142/S0252959902000158.  Google Scholar

[3]

C. Cancès, M. Ibrahim and M. Saad, A Nonlinear CVFE Scheme for an anisotropic degenerate nonlinear Keller-Segel model, European Consortium for Mathematics in Industry, 22 (2014), 1037-1046.  doi: 10.1007/978-3-319-23413-7_145.  Google Scholar

[4]

G. Chamoun, M. Ibrahim, M. Saad and R. Talhouk, Numerical simulation of heterogeneous steady states for a reaction-diffusion degenerate Keller-Segel model, European Consortium for Mathematics in Industry, 30 (2019), 411-417.  doi: 10.1007/978-3-030-27550-1_52.  Google Scholar

[5]

G. Chamoun, M. Saad and R. Talhouk, A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion, Comput. Math. Appl., 68 (2014), 1052-1070.  doi: 10.1016/j.camwa.2014.04.010.  Google Scholar

[6]

G. Chamoun, M. Saad and R. Talhouk, Monotone combined edge finite volume–finite element scheme for anisotropic Keller-Segel model, Numer. Methods Partial Differential Equations, 30 (2014), 1030-1065.  doi: 10.1002/num.21858.  Google Scholar

[7]

M. H. Cohen and A. Robertson, Chemotaxis and the early stages of aggregation in cellular slime molds, Journal of Theoretical Biology, 31 (1971), 119-130.  doi: 10.1016/0022-5193(71)90125-1.  Google Scholar

[8]

P. De Leenheer, J. Gopalakrishnan and E. Zuhr, Instability in a generalized Keller-Segel model, Journal of Biological Dynamics, 6 (2012), 974-991.  doi: 10.1080/17513758.2012.714478.  Google Scholar

[9]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of Numerical Analysis, 7 (2000), 713-1020.   Google Scholar

[10]

S. Fu and F. Cao, Pattern formation of a Keller-Segel model with the source term $u^p(1-u)$, J. Math., (2013), Art. ID 454513, 11 pp. doi: 10.1155/2013/454513.  Google Scholar

[11]

D. F. Griffiths and D. J. Higham, Numerical Methods For Ordinary Differential Equations. Initial Value Problems, Springer-Verlag London, Ltd., London, 2010. doi: 10.1007/978-0-85729-148-6.  Google Scholar

[12]

Y. Guo, Instability of symmetric vortices with large charge and coupling constant, Comm. Pure Appl. Math., 49 (1996), 1051-1080.  doi: 10.1002/(SICI)1097-0312(199610)49:10<1051::AID-CPA2>3.0.CO;2-D.  Google Scholar

[13]

Y. Guo, C. Hallstrom and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys., 245 (2004), 297-354.  doi: 10.1007/s00220-003-1017-z.  Google Scholar

[14]

Y. Guo and H. J. Hwang, Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.  doi: 10.1016/j.jde.2010.07.025.  Google Scholar

[15]

Y. Guo and H. J. Hwang, Pattern formation. (Ⅱ). The Turing instability, Proc. Amer. Math. Soc., 135 (2007), 2855-2866.  doi: 10.1090/S0002-9939-07-08850-8.  Google Scholar

[16]

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

[17]

T. Hoang and H. J. Hwang, Turing instability in a general system, Nonlinear Anal., 91 (2013), 93-113.  doi: 10.1016/j.na.2013.06.010.  Google Scholar

[18]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.   Google Scholar

[19]

M. Ibrahim and M. Saad, On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model, Comput. Math. Appl., 68 (2014), 1032-1051.  doi: 10.1016/j.camwa.2014.03.010.  Google Scholar

[20]

H.-Y. Jin and Z.-A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[21]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[22]

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

[23]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. Google Scholar

[24]

P. Laurençcot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, Nonlinear Elliptic and Parabolic Problems, 64 (2005), 273-290.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar

[25]

R. J. LeVeque, Conservative methods for nonlinear problems, in Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1990,122–135. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[26]

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

[27]

P. K. Maini, The impact of Turing's work on pattern formation in biology, Mathematics Today, 40 (2004), 140-141.   Google Scholar

[28]

J. D. Murray, Mathematical biology Ⅱ: Spatial models and biomedical applications, in Interdisciplinary Applied Mathematics, vol. 18, Springer-Verlag, New York, 2003.  Google Scholar

[29]

E. Sander and T. Wanner, Pattern formation in a nonlinear model for animal coats, J. Differential Equations, 191 (2003), 143-174.  doi: 10.1016/S0022-0396(02)00156-0.  Google Scholar

[30]

Y. 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

[31]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.   Google Scholar

[32]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.  Google Scholar

[33]

S. Wu, J. Shi and B. Wu, Global existence of solutions to an attraction-repulsion chemotaxis model with growth, Commun. Pure Appl. Anal., 16 (2017), 1037-1058.  doi: 10.3934/cpaa.2017050.  Google Scholar

Figure 1.  Unstructured triangular mesh for the space domain $ {\Omega} = {\left({0,1}\right)}\times{\left({0,1}\right)} $ with 14336 acute angle triangles
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Figure 2.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the chemosensitivity strength $ \zeta $ increases beyond the critical value $ \zeta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi
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Figure 3.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $
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Figure 4.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero. 2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)
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Figure 5.  First row from left to right: Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 2.5 $, $ t = 325 $, and $ t = 997.5 $. Second row from left to right: Evolution of the heterogeneous stationary solutions at the same moments as for the evolution of $ u{\left({ {\mathbf{x}},t}\right)} $
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Figure 6.  Similarities of patterns between the nonlinear evolution $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)
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Figure 7.  Time evolution of the difference in $ {L^{2}} $ between $ u{\left({ {\mathbf{x}},t}\right)} $ and the heterogeneous solution
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Figure 8.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the death rate $ \beta $ decreases below the critical value $ \beta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi, and pattern formation can be expected
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Figure 9.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $
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Figure 10.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero.2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)
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Figure 11.  First row from left to right. Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 10 $, $ t = 70 $, and $ t = 750 $. Second row from left to right. Evolution of the heterogeneous stationary solutions at the same moments as for $ u{\left({ {\mathbf{x}},t}\right)} $
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Figure 12.  Similarities of patterns between the nonlinear evolution $ u({\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)
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