October  2018, 23(8): 3023-3045. doi: 10.3934/dcdsb.2017199

Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

1. 

Dipartimento di Matematica e Informatica, Università di Cagliari, V. le Merello 92,09123. Cagliari, Italy

2. 

Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, United Kingdom

* Corresponding author: Giuseppe Viglialoro

Received  December 2016 Revised  April 2017 Published  July 2017

Fund Project: TEW would like to thank St John's College, Oxford and the Mathematical Biosciences Institute (MBI) at Ohio State University, for financially supporting this research through the National Science Foundation grant DMS 1440386 and BBSRC grant BKNXBKOO BK00.16

In this paper we study the chemotaxis-system
$\begin{equation*}\begin{cases}u_{t}=Δ u-χ \nabla · (u\nabla v)+g(u)&x∈ Ω, t>0, \\v_{t}=Δ v-v+u&x∈ Ω, t>0,\end{cases}\end{equation*}$
defined in a convex smooth and bounded domain
$Ω$
of
$\mathbb{R}^n$
,
$n≥ 1$
, with
$χ>0$
and endowed with homogeneous Neumann boundary conditions. The source
$g$
behaves similarly to the logistic function and satisfies
$g(s)≤ a -bs^α$
, for
$s≥ 0$
, with
$a≥ 0$
,
$b>0$
and
$α>1$
. Continuing the research initiated in [33], where for appropriate
$1 < p < α < 2$
and
$(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$
the global existence of very weak solutions
$(u,v)$
to the system (for any
$n≥ 1$
) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when
$n=3$
, we establish that
-for all
$τ>0$
an upper bound for
$\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$
can be prescribed in a such a way that
$(u,v)$
is bounded and Hölder continuous beyond
$τ$
;
-for all
$(u_0,v_0)$
, and sufficiently small ratio
$\frac{a}{b}$
, there exists a
$T>0$
such that
$(u,v)$
is bounded and Hölder continuous beyond
$T$
.
Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.
Citation: Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199
References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc., 74 (2006), 453-474.  doi: 10.1112/S0024610706023015.  Google Scholar

[2]

J. L. Aragón, R. A. Barrio, T. E. Woolley, R. E. Baker and P. K. Maini, Nonlinear effects on turing patterns: Time oscillations and chaos, Phys. Rev. E, 86 (2012), 026201. Google Scholar

[3]

N. BellomoA. BellouquidY. 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.  Google Scholar

[4]

J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theor. Biol., 334 (2013), 1 – 12, URL http://www.sciencedirect.com/science/article/pii/S0022519313002646. doi: 10.1016/j.jtbi.2013.05.033.  Google Scholar

[5]

S. W. ChoS. KwakT. E. WoolleyM. J. LeeE. J. KimR. E. BakerH. J. KimJ. S. ShinC. TickleP. K. Maini and H. S. Jung, Interactions between shh, sostdc1 and wnt signaling and a new feedback loop for spatial patterning of the teeth, Development, 138 (2011), 1807-1816.  doi: 10.1242/dev.056051.  Google Scholar

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T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.  Google Scholar

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M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Discret. Contin. Dyn. Syst. Suppl, 409–417. doi: 10.3934/proc.2015.0409.  Google Scholar

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D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Eqns., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, T. Am. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

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

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type in Translations of Mathematical Monographs, vol. 23, American Mathematical Society, 1988.  Google Scholar

[13]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equations., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[14]

P. K. MainiT. E. WoolleyR. E. BakerE. A. Gaffney and S. S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.  Google Scholar

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P. K. Maini, T. E. Woolley, E. A. Gaffney and R. E. Baker, The Once and Future Turing chapter 15: Biological pattern formation, Cambridge University Press, 2016. Google Scholar

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M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolicparabolic Keller–Segel system, Discret. Contin. Dyn. Syst. Suppl, 809–916. doi: 10.3934/proc.2015.0809.  Google Scholar

[17]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Method. Appl. Sci., 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar

[18]

M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulg. Sci., 69 (2016), 687-696.   Google Scholar

[19]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications vol. 2, 3rd edition, Springer-Verlag, 2003.  Google Scholar

[20]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[21]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[22]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvacioj., 44 (2001), 441-470.   Google Scholar

[23]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676.  doi: 10.1016/j.jmaa.2011.06.086.  Google Scholar

[25]

L. -E. Persson and N. Samko, Inequalities and Convexity, in Operator Theory, Operator Algebras and Applications, Springer Basel, 2014,279–306. doi: 10.1007/978-3-0348-0816-3_17.  Google Scholar

[26]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[27]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.  Google Scholar

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Eqns., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[29]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Part. Diff. Eq., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[30]

P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121.   Google Scholar

[31]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232.   Google Scholar

[32]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Int. Eqns., 29 (2016), 359-376.   Google Scholar

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.  Google Scholar

[35]

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.  Google Scholar

[36]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[37]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part. Diff. Eq., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[38]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Method. Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[39]

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.  Google Scholar

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[41]

T. E. Woolley, Spatiotemporal Behaviour of Stochastic and Continuum Models for Biological Signalling on Stationary and Growing Domains} PhD thesis, University of Oxford, 2011. Google Scholar

[42]

T. E. Woolley, 50 Visions of Mathematics chapter 48: Mighty Morphogenesis, Oxford Univ. Press, 2014. Google Scholar

[43]

T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini, Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation Phys. Rev. E, 84 (2011), 046216. doi: 10.1103/PhysRevE.84.046216.  Google Scholar

[44]

T. E. WoolleyR. E. BakerC. TickleP. K. Maini and M. Towers, Mathematical modelling of digit specification by a sonic hedgehog gradient, Dev. Dynam., 243 (2014), 290-298.  doi: 10.1002/dvdy.24068.  Google Scholar

show all references

References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc., 74 (2006), 453-474.  doi: 10.1112/S0024610706023015.  Google Scholar

[2]

J. L. Aragón, R. A. Barrio, T. E. Woolley, R. E. Baker and P. K. Maini, Nonlinear effects on turing patterns: Time oscillations and chaos, Phys. Rev. E, 86 (2012), 026201. Google Scholar

[3]

N. BellomoA. BellouquidY. 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.  Google Scholar

[4]

J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theor. Biol., 334 (2013), 1 – 12, URL http://www.sciencedirect.com/science/article/pii/S0022519313002646. doi: 10.1016/j.jtbi.2013.05.033.  Google Scholar

[5]

S. W. ChoS. KwakT. E. WoolleyM. J. LeeE. J. KimR. E. BakerH. J. KimJ. S. ShinC. TickleP. K. Maini and H. S. Jung, Interactions between shh, sostdc1 and wnt signaling and a new feedback loop for spatial patterning of the teeth, Development, 138 (2011), 1807-1816.  doi: 10.1242/dev.056051.  Google Scholar

[6]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[7]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Discret. Contin. Dyn. Syst. Suppl, 409–417. doi: 10.3934/proc.2015.0409.  Google Scholar

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[9]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Eqns., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, T. Am. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

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

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type in Translations of Mathematical Monographs, vol. 23, American Mathematical Society, 1988.  Google Scholar

[13]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equations., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[14]

P. K. MainiT. E. WoolleyR. E. BakerE. A. Gaffney and S. S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.  Google Scholar

[15]

P. K. Maini, T. E. Woolley, E. A. Gaffney and R. E. Baker, The Once and Future Turing chapter 15: Biological pattern formation, Cambridge University Press, 2016. Google Scholar

[16]

M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolicparabolic Keller–Segel system, Discret. Contin. Dyn. Syst. Suppl, 809–916. doi: 10.3934/proc.2015.0809.  Google Scholar

[17]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Method. Appl. Sci., 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar

[18]

M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulg. Sci., 69 (2016), 687-696.   Google Scholar

[19]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications vol. 2, 3rd edition, Springer-Verlag, 2003.  Google Scholar

[20]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[21]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[22]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvacioj., 44 (2001), 441-470.   Google Scholar

[23]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676.  doi: 10.1016/j.jmaa.2011.06.086.  Google Scholar

[25]

L. -E. Persson and N. Samko, Inequalities and Convexity, in Operator Theory, Operator Algebras and Applications, Springer Basel, 2014,279–306. doi: 10.1007/978-3-0348-0816-3_17.  Google Scholar

[26]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[27]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.  Google Scholar

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Eqns., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[29]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Part. Diff. Eq., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[30]

P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121.   Google Scholar

[31]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232.   Google Scholar

[32]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Int. Eqns., 29 (2016), 359-376.   Google Scholar

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.  Google Scholar

[35]

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.  Google Scholar

[36]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[37]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part. Diff. Eq., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[38]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Method. Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[39]

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.  Google Scholar

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[41]

T. E. Woolley, Spatiotemporal Behaviour of Stochastic and Continuum Models for Biological Signalling on Stationary and Growing Domains} PhD thesis, University of Oxford, 2011. Google Scholar

[42]

T. E. Woolley, 50 Visions of Mathematics chapter 48: Mighty Morphogenesis, Oxford Univ. Press, 2014. Google Scholar

[43]

T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini, Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation Phys. Rev. E, 84 (2011), 046216. doi: 10.1103/PhysRevE.84.046216.  Google Scholar

[44]

T. E. WoolleyR. E. BakerC. TickleP. K. Maini and M. Towers, Mathematical modelling of digit specification by a sonic hedgehog gradient, Dev. Dynam., 243 (2014), 290-298.  doi: 10.1002/dvdy.24068.  Google Scholar

Figure 1.  Simulations of system (45) in one dimension with varying value of $\alpha$, given beneath each subfigure. Each subfigure contains the system evaluated at the time points $t=1$, 10, 50 and 100. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1000 equally spaced points
Figure 2.  Simulations of system (45) in one dimension. The simulations are nearly identical to those seen in Figure 1(a). However, each simulation involves a single parameter change. Specifically, in (a) a larger initial condition for $u$ was used (100 was added to the mean); in (b) the parameter $b$ was reduced to 0.2; Finally, in (c) the spatial solution domain has been reduced from 10 to 1
Figure 3.  Simulations of system (45) in two dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was triangulated into 24, 968 finite elements. The figure inset of (b) shows the full extent of the peak, which is growing without bound
Figure 4.  Simulations of system (45) illustrating the density of $u$ in three dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1, 139, 254 voxel elements. Apart from the light grey ball illustrating the boundary of the solution domain the images illustrate isosurfaces of the solution (i.e. surface that represent points of a constant value, thus, they are the three-dimensional analogue of contours). In Figure (a) there are five isosurfaces of value 1, 1.25, 1.5 1.75 and 2, coloured, yellow, green, blue, red and black, respectively. In Figure (b) there are three isosurfaces of value 1, 10, and $10^6$, coloured, yellow, blue and black, respectively
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