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February  2020, 13(2): 279-292. doi: 10.3934/dcdss.2020016

On a Parabolic-ODE system of chemotaxis

Mihaela Negreanu 1, and  J. Ignacio Tello 2,3,,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2. 

Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid 28031, Spain
3. 

Center for Computation and Simulation, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain

† Corresponding author

Received  May 2017 Revised  April 2018 Published  January 2019

Fund Project: This work has been supported by the Project MTM2017-83391-P from MICINN (Spain).

In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "
$u$
" which moves towards a higher concentration of a chemical species "
$v$
" in a bounded domain of
$ \mathbb{R}^n$
. The chemical "
$v$
" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation
$v_t = h(u, v).$
We obtain results concerning the bifurcation of constant steady states under the assumption
$ h_v+χ u h_u>0 $
with growth terms
$g$
. The Parabolic-ODE problem is also considered for the case
$h_v+χ u h_u = 0$
 without growth terms, i.e.
$g \equiv 0$
. Global existence of solutions is obtained for a range of initial data.
Keywords: Parabolic-ODE, chemotaxis, global existence.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 279-292. doi: 10.3934/dcdss.2020016
References:
[1]

A. R. A. Anderson and M. A. I. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.

[2]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete& Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.

[3]

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.  doi: 10.1093/imamat/hxw036.

[4]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in Epithelia, Physical Review E Stat Nonlin Soft Matter Phys, 75 (2007), 011901. doi: 10.1103/PhysRevE.75.011901.

[5]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[6]

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.

[7]

L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia 2005. doi: 10.1137/1.9780898719147.

[8]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533.  doi: 10.1142/S0218202504003337.

[9]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[10]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.  doi: 10.1016/S0022-247X(02)00147-6.

[11]

M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a twodimensional two-species chemotaxis-Navier- Stokes system with competitive kinetics, Journal of Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math. Ver., 105 (2003), 103-165. 

[13]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.

[14]

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.

[15]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[16]

H. Kielhöfer, Bifurcation Theory, Springer, 2004. doi: 10.1007/b97365.

[17]

P. Krzyżanowski, P. Laurençot and D. Wrzosek, Mathematical models of receptor-mediated transport of morphogens, M3AS, 20 (2010), 2021-2052.  doi: 10.1142/S0218202510004866.

[18]

A. Kubo, H. Hoshino and K. Kimura, Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion, Proceedings of 10th AIMS Conference, (2015), 733-744.  doi: 10.3934/proc.2015.0733.

[19]

A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55.  doi: 10.1016/j.cam.2006.04.027.

[20]

A. Kubo and J. I. Tello, Mathematical analysis of a model of chemotaxis with competition terms, Differential and Integral Equations, 29 (2016), 441-454. 

[21]

D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185.  doi: 10.1007/BF02540222.

[22]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[23]

H. A. Levine and J. Renclawowicz, Singularity formation in chemotaxis - a conjecture of nagai, SIAM J. Appl. Math., 65 (2004), 336-360.  doi: 10.1137/S0036139903431725.

[24]

H. A. Levine, B. P. Sleeman and N. Nilsen-Hamilton, A mathematical modeling for the roles of pericytes and macrophages in the initiation of angiogenesis I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 75-115.  doi: 10.1016/S0025-5564(00)00034-1.

[25]

Y. Li, K. Lin and C. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxishaptotaxis model in high dimensions, Applied Mathematics Letters, 50 (2015), 91-97.  doi: 10.1016/j.aml.2015.06.010.

[26]

M. Malogrosz, Well-posedness and asymptotic behavior of a multidimensonal model of morphogen transport, J. Evol. Eq., 12 (2012), 353-366.  doi: 10.1007/s00028-012-0135-5.

[27]

M. Małogrosz, A model of morphogen transport in the presence of glypicans II, Journal of Mathematical Analysis and Applications, 433 (2016), 642-680.  doi: 10.1016/j.jmaa.2015.07.053.

[28]

J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773.  doi: 10.1088/0951-7715/18/6/018.

[29]

J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17.  doi: 10.1007/s00285-004-0308-0.

[30]

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

[31]

A.I. Muñoz and J. I. Tello, Mathematical analysis and numerical simulation of a model of morphogenesis, Mathematical Biosciences and Engineering, 8 (2011), 1035-1059.  doi: 10.3934/mbe.2011.8.1035.

[32]

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.

[33]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 2669-2688.  doi: 10.3934/dcdsb.2013.18.2669.

[34]

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.

[35]

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.

[36]

C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407.

[37]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC$^{\prime}$s of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. 

[38]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods Appl. Sci., 24 (2001), 405-426.  doi: 10.1002/mma.212.

[39]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.1137/S0036139902415117.

[40]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochactic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.  doi: 10.1137/S0036139998342065.

[41]

C. Stinner, J. I. Tello and M. Winkler, Mathematical analysis of a model of chemotaxis arising from morphogenesis, Math. Methods Appl. Sci., 35 (2012), 445-465.  doi: 10.1002/mma.1573.

[42]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.

[43]

T. Suzuki, Mathematical models of tumor growth systems, Mathematica Bohemica, 137 (2012), 201-218. 

[44]

T. Suzuki and R. Takahashi, Global in time solution to a class of tumour growth systems, Adv. Math. Sci. Appl., 19 (2009), 503-524. 

[45]

J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343-361.  doi: 10.3934/dcds.2009.25.343.

[46]

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.

[47]

A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[48]

X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531.  doi: 10.1090/qam/1914439.

show all references

References:
[1]

A. R. A. Anderson and M. A. I. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.

[2]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete& Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.

[3]

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.  doi: 10.1093/imamat/hxw036.

[4]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in Epithelia, Physical Review E Stat Nonlin Soft Matter Phys, 75 (2007), 011901. doi: 10.1103/PhysRevE.75.011901.

[5]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[6]

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.

[7]

L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia 2005. doi: 10.1137/1.9780898719147.

[8]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533.  doi: 10.1142/S0218202504003337.

[9]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[10]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.  doi: 10.1016/S0022-247X(02)00147-6.

[11]

M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a twodimensional two-species chemotaxis-Navier- Stokes system with competitive kinetics, Journal of Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math. Ver., 105 (2003), 103-165. 

[13]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.

[14]

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.

[15]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[16]

H. Kielhöfer, Bifurcation Theory, Springer, 2004. doi: 10.1007/b97365.

[17]

P. Krzyżanowski, P. Laurençot and D. Wrzosek, Mathematical models of receptor-mediated transport of morphogens, M3AS, 20 (2010), 2021-2052.  doi: 10.1142/S0218202510004866.

[18]

A. Kubo, H. Hoshino and K. Kimura, Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion, Proceedings of 10th AIMS Conference, (2015), 733-744.  doi: 10.3934/proc.2015.0733.

[19]

A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55.  doi: 10.1016/j.cam.2006.04.027.

[20]

A. Kubo and J. I. Tello, Mathematical analysis of a model of chemotaxis with competition terms, Differential and Integral Equations, 29 (2016), 441-454. 

[21]

D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185.  doi: 10.1007/BF02540222.

[22]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[23]

H. A. Levine and J. Renclawowicz, Singularity formation in chemotaxis - a conjecture of nagai, SIAM J. Appl. Math., 65 (2004), 336-360.  doi: 10.1137/S0036139903431725.

[24]

H. A. Levine, B. P. Sleeman and N. Nilsen-Hamilton, A mathematical modeling for the roles of pericytes and macrophages in the initiation of angiogenesis I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 75-115.  doi: 10.1016/S0025-5564(00)00034-1.

[25]

Y. Li, K. Lin and C. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxishaptotaxis model in high dimensions, Applied Mathematics Letters, 50 (2015), 91-97.  doi: 10.1016/j.aml.2015.06.010.

[26]

M. Malogrosz, Well-posedness and asymptotic behavior of a multidimensonal model of morphogen transport, J. Evol. Eq., 12 (2012), 353-366.  doi: 10.1007/s00028-012-0135-5.

[27]

M. Małogrosz, A model of morphogen transport in the presence of glypicans II, Journal of Mathematical Analysis and Applications, 433 (2016), 642-680.  doi: 10.1016/j.jmaa.2015.07.053.

[28]

J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773.  doi: 10.1088/0951-7715/18/6/018.

[29]

J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17.  doi: 10.1007/s00285-004-0308-0.

[30]

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

[31]

A.I. Muñoz and J. I. Tello, Mathematical analysis and numerical simulation of a model of morphogenesis, Mathematical Biosciences and Engineering, 8 (2011), 1035-1059.  doi: 10.3934/mbe.2011.8.1035.

[32]

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.

[33]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 2669-2688.  doi: 10.3934/dcdsb.2013.18.2669.

[34]

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.

[35]

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.

[36]

C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407.

[37]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC$^{\prime}$s of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. 

[38]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods Appl. Sci., 24 (2001), 405-426.  doi: 10.1002/mma.212.

[39]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.1137/S0036139902415117.

[40]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochactic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.  doi: 10.1137/S0036139998342065.

[41]

C. Stinner, J. I. Tello and M. Winkler, Mathematical analysis of a model of chemotaxis arising from morphogenesis, Math. Methods Appl. Sci., 35 (2012), 445-465.  doi: 10.1002/mma.1573.

[42]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.

[43]

T. Suzuki, Mathematical models of tumor growth systems, Mathematica Bohemica, 137 (2012), 201-218. 

[44]

T. Suzuki and R. Takahashi, Global in time solution to a class of tumour growth systems, Adv. Math. Sci. Appl., 19 (2009), 503-524. 

[45]

J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343-361.  doi: 10.3934/dcds.2009.25.343.

[46]

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.

[47]

A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[48]

X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531.  doi: 10.1090/qam/1914439.

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