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

On a Parabolic-ODE system of chemotaxis

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.
Citation: Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 279-292. doi: 10.3934/dcdss.2020016
References:
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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.  Google Scholar

[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.  Google Scholar

[3]

T. BlackJ. 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.  Google Scholar

[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.  Google Scholar

[5]

N. BellomoA. BellouquidY. 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.  Google Scholar

[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.  Google Scholar

[7]

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

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

[9]

M. A. FontelosA. 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.  Google Scholar

[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.  Google Scholar

[11]

M. HirataS. KurimaM. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

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A. KuboH. 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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

H. A. LevineB. 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.  Google Scholar

[25]

Y. LiK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. H. MerkingD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

C. StinnerJ. 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.  Google Scholar

[42]

C. StinnerJ. 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.  Google Scholar

[43]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. BlackJ. 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.  Google Scholar

[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.  Google Scholar

[5]

N. BellomoA. BellouquidY. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

M. A. FontelosA. 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.  Google Scholar

[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.  Google Scholar

[11]

M. HirataS. KurimaM. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

A. KuboH. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

H. A. LevineB. 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.  Google Scholar

[25]

Y. LiK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. H. MerkingD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

C. StinnerJ. 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.  Google Scholar

[42]

C. StinnerJ. 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.  Google Scholar

[43]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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