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April  2022, 21(4): 1447-1479. doi: 10.3934/cpaa.2022025

On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops

1. 

School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401131, China

2. 

Department of Mathematics, University of Miami, Coral Gables, FL33146, USA

3. 

Sylvester Comprehensive Cancer Center, University of Miami Miller School of Medicine, Miami, FL 33136, USA

*Corresponding author

Received  July 2021 Revised  November 2021 Published  April 2022 Early access  February 2022

Fund Project: Research was partially supported by Chongqing Science and Technology Commission Project (No. sctc2020jcyj-msxmX0560), Research Project of Chongqing Education Commission (No. CXQT21014), China Postdoctoral Science Foundation (No. 2017M622990, No. 2018T110956), and NSFC (No.11701461, No. 11771168)

In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops:
$ \begin{equation*} \label{IBVP} \left\{ \begin{aligned} &u_t = \nabla\cdot(D_1(u)\nabla u-S_1(u)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_1 v_t = \Delta v- v+w, &\qquad x\in\Omega, \, t>0, \\ &w_t = \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_2 z_t = \Delta z- z+ u, &\qquad x\in\Omega, \, t>0, \end{aligned} \right. \end{equation*} $
where
$ u(t, x) $
and
$ w(t, x) $
denote the density of macrophages and tumor cells at time
$ t $
and location
$ x\in \Omega, $
respectively,
$ v(t, x) $
and
$ z(t, x) $
represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time
$ t $
and location
$ x\in \Omega, $
respectively.
$ \Omega\subset \mathbb{R}^n $
is a bounded region with smooth boundary,
$ \tau_i\ge 0 \; (i = 1, 2) $
,
$ D_i(s)\ge d_i(s+1)^{m_i-1} $
with parameters
$ m_i\ge 1 \; (i = 1, 2) $
and
$ S_j(s)\lesssim (s+1)^{q_j} $
with parameters
$ q_j>0 \;(j = 1, 2, 3) $
. For the case without autocrine loop (i.e.,
$ S_3(w) = 0 $
), it is shown that when
$ q_j\le 1 \; (j = 1, 2) $
, if one of
$ q_j $
is smaller than one or one of
$ m_i $
is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when
$ m_1 = m_2 = q_1 = q_2 = 1 $
, an inequality involving the product
$ d_1d_2 $
and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of
$ d_i $
to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e
$ S_3(w)\ne 0 $
), similar results hold only if
$ q_3<1 $
. If
$ q_3 = 1 $
, solutions to the IBVP exist globally only when
$ d_2 $
is suitably large or the mass of species
$ w $
is suitably small.
Citation: Li Xie, Shigui Ruan. On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1447-1479. doi: 10.3934/cpaa.2022025
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

A. H. BeckI. EspinosaB. EdrisR. LiK. MontgomeryS. ZhuS. VarmaR.J. MarinelliM. van de Rijn and R. B. West, The macrophage colony-stimulating factor response signature in breast carcinoma, Clin. Cancer Res., 15 (2009), 778-787. 

[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.

[4]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci.Appl., 9 (1999), 347-359. 

[5]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[6]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[8]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[9]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[10]

C. ConcaE. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Eur. J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258.

[11]

E. Espejo ArenasA. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029.

[12]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[13]

A. S. Harney, Real-time imaging reveals local, transient vascular permeability, and tumor cell intravasation stimulated by TIE2hi macrophage-derived VEGFA, Cancer Discovery, 5 (2015), 932-943. 

[14]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. 

[15]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[16]

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

[17]

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.

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[19]

H. KnútsdóttirE. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.

[20]

E.Y. LinV. Gouon-EvansA. V. Nguyen and J. W. Pollard, The macrophage growth factor CSF-1 in mammary gland development and tumor progression, J. Mammary Gland Biol. Neoplasia, 7 (2002), 147-162. 

[21]

K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var., 59 (2020), 35 pp. doi: 10.1007/s00526-020-01777-7.

[22]

C. MuL. WangP. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022.

[23]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[24]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[25]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[26]

A. PatsialouJ. WyckoffY. WangS. GoswamiE. R. Stanley and J. S. Condeelis, Invasion of human breast cancer cells in vivo requires both paracrine and autocrine loops involving the colony-stimulating factor-1 receptor, Cancer Res., 69 (2009), 9498-9506. 

[27]

K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. 

[28]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.

[29]

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

[30]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[31]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[32]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[33]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[34]

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

[35]

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

[36]

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

[37]

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.

[38]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.

[39]

J. B. Wyckoff, Direct visualization of macrophage-assisted tumor cell intravasation in mammary tumors, Cancer Res., 67 (2007), 2649-2656. 

[40]

L. Xie, On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.  doi: 10.1016/j.nonrwa.2019.02.005.

[41]

L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.

[42]

L. Xie and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.

[43]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differ. Equ., 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032.

[44]

C. YangX. CaoZ. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[45]

H. YuW. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9.

[46]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[47]

J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/tmna.2016.082.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

A. H. BeckI. EspinosaB. EdrisR. LiK. MontgomeryS. ZhuS. VarmaR.J. MarinelliM. van de Rijn and R. B. West, The macrophage colony-stimulating factor response signature in breast carcinoma, Clin. Cancer Res., 15 (2009), 778-787. 

[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.

[4]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci.Appl., 9 (1999), 347-359. 

[5]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[6]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[8]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[9]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[10]

C. ConcaE. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Eur. J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258.

[11]

E. Espejo ArenasA. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029.

[12]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[13]

A. S. Harney, Real-time imaging reveals local, transient vascular permeability, and tumor cell intravasation stimulated by TIE2hi macrophage-derived VEGFA, Cancer Discovery, 5 (2015), 932-943. 

[14]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. 

[15]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[16]

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

[17]

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.

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[19]

H. KnútsdóttirE. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.

[20]

E.Y. LinV. Gouon-EvansA. V. Nguyen and J. W. Pollard, The macrophage growth factor CSF-1 in mammary gland development and tumor progression, J. Mammary Gland Biol. Neoplasia, 7 (2002), 147-162. 

[21]

K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var., 59 (2020), 35 pp. doi: 10.1007/s00526-020-01777-7.

[22]

C. MuL. WangP. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022.

[23]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[24]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[25]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[26]

A. PatsialouJ. WyckoffY. WangS. GoswamiE. R. Stanley and J. S. Condeelis, Invasion of human breast cancer cells in vivo requires both paracrine and autocrine loops involving the colony-stimulating factor-1 receptor, Cancer Res., 69 (2009), 9498-9506. 

[27]

K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. 

[28]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.

[29]

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

[30]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[31]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[32]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[33]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[34]

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

[35]

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

[36]

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

[37]

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.

[38]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.

[39]

J. B. Wyckoff, Direct visualization of macrophage-assisted tumor cell intravasation in mammary tumors, Cancer Res., 67 (2007), 2649-2656. 

[40]

L. Xie, On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.  doi: 10.1016/j.nonrwa.2019.02.005.

[41]

L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.

[42]

L. Xie and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.

[43]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differ. Equ., 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032.

[44]

C. YangX. CaoZ. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[45]

H. YuW. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9.

[46]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[47]

J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/tmna.2016.082.

Figure 1.  Schematic of EGF/CSF-1 paracrine and autocrine interactions between tumor cells and macrophages. Macrophages secrete EGF and have CSF-1 receptors. Tumor cells secrete CSF-1 and have both EGF/CSF-1 receptors. When CSF-1R on macrophages are activated, the macrophages respond by secreting EGF and chemotact in the direction of the CSF-1 gradient. When EGFR on tumor cells are activated, the tumor cells respond by secreting CSF-1 and chemotact up the EGF gradient. This paracrine signaling loop enables tumor cells to migrate alongside macrophages away from the primary tumor. Moreover, the tumor cells not only secrete CSF-1 but also respond to the concentration gradient of CSF-1 and this is the CSF-1/CSF-1R autocrine signalling loop (Patsialou et al. [26]). EGFR$ = $EGF receptors, CSF-1R$ = $CSF-1 receptors
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