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September  2017, 22(7): 2777-2793. doi: 10.3934/dcdsb.2017135

Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems

Universität Paderborn, Institut für Mathematik, Warburger Str. 100,33098 Paderborn, Germany

Received  August 2016 Revised  September 2016 Published  April 2017

Fund Project: The author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks

We consider the no-flux initial-boundary value problem for Keller-Segel-type chemotaxis growth systems of the form
$\begin{eqnarray*} ≤\left\{ \begin{array}{ll} u_t=Δ u -χ \nabla · (u\nabla v) + ρ u -μ u^2, & x∈Ω, \ t>0, \\ v_t=Δ v -v+u, & x∈Ω, \ t>0, \end{array} \right. \end{eqnarray*}$
in a ball
$Ω\subset\mathbb{R}^n$
,
$n≥ 3$
, with parameters
$χ>0, ρ≥ 0$
and
$μ>0$
.
By means of an argument based on a conditional quasi-energy inequality, it is firstly shown that if
$χ=1$
is fixed, then for any given
$K>0$
and
$T>0$
one can find radially symmetric initial data, possibly depending on
$K$
and
$T$
, such that for arbitrary
$μ∈ (0, 1)$
the corresponding local-in-time classical solution
$(u, v)$
satisfies
$\begin{eqnarray*} u(x, t) > \frac{K}{μ} \end{eqnarray*}$
with some
$x∈Ω$
and
$t∈ (0, T)$
; in fact, this growth phenomenon is actually identified as being generic in the sense that the set of all initial data having this property is dense in the set of all suitably regular radial initial data in a certain topology.
Secondly, turning a focus on possible effects of large chemotactic sensitivities, on the basis of the above it is shown that when
$ρ≥ 0$
and
$μ>0$
are fixed, then for all
$L>0, T>0$
and
$χ>μ$
one can fix radial initial data
$(u_{0, χ}, v_{0, χ})$
which decay in
$L^∞(Ω)× W^{1, ∞}(Ω)$
as
$χ\to∞$
, and which are such that for the respective solution
$(u_χ, v_χ)$
there exist
$x∈Ω$
and
$t∈ (0, T)$
fulfilling
$\begin{eqnarray*} u_χ(x, t) > L. \end{eqnarray*}$
Citation: Michael Winkler. Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2777-2793. doi: 10.3934/dcdsb.2017135
References:
[1]

M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal Real World Appl., 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011. Google Scholar

[2]

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

[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]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947. Google Scholar

[5]

H. J. EberlD. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-175. Google Scholar

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scu. Norm. Super. Pisa Cl. Sci., 24 (1997), 663-683. Google Scholar

[7]

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

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte DMV, 105 (2003), 103-165. Google Scholar

[9]

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

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Google Scholar

[11]

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

[12]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discr. Cont. Dyn. Syst. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499. Google Scholar

[13]

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

[14]

G. MeralC. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discr. Cont. Dyn. Syst. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189. Google Scholar

[15]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar

[16]

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. Google Scholar

[17]

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

[18]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. Google Scholar

[19]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. Google Scholar

[20]

K. J. PainterP. K. Maini and H. G. Othmer, Complex spatial patterns in a hybrid chemotaxis reaction-diffusion model, J. Math. Biol., 41 (2000), 285-314. Google Scholar

[21]

C. StinnerC. 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. Google Scholar

[22]

Z. SzymańskaC. Morales RodrigoM. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Mod. Meth. Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425. Google Scholar

[23]

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

[24]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Eq., 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019. Google Scholar

[25]

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

[26]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767, arXiv: 1112.4156v1 doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[27]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Eq., 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar

[28]

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. Google Scholar

show all references

References:
[1]

M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal Real World Appl., 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011. Google Scholar

[2]

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

[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]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947. Google Scholar

[5]

H. J. EberlD. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-175. Google Scholar

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scu. Norm. Super. Pisa Cl. Sci., 24 (1997), 663-683. Google Scholar

[7]

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

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte DMV, 105 (2003), 103-165. Google Scholar

[9]

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

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Google Scholar

[11]

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

[12]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discr. Cont. Dyn. Syst. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499. Google Scholar

[13]

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

[14]

G. MeralC. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discr. Cont. Dyn. Syst. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189. Google Scholar

[15]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar

[16]

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. Google Scholar

[17]

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

[18]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. Google Scholar

[19]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. Google Scholar

[20]

K. J. PainterP. K. Maini and H. G. Othmer, Complex spatial patterns in a hybrid chemotaxis reaction-diffusion model, J. Math. Biol., 41 (2000), 285-314. Google Scholar

[21]

C. StinnerC. 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. Google Scholar

[22]

Z. SzymańskaC. Morales RodrigoM. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Mod. Meth. Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425. Google Scholar

[23]

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

[24]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Eq., 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019. Google Scholar

[25]

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

[26]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767, arXiv: 1112.4156v1 doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[27]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Eq., 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar

[28]

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. Google Scholar

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