October  2019, 24(10): 5461-5480. doi: 10.3934/dcdsb.2019066

Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics

School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

* Corresponding author

Received  June 2018 Revised  November 2018 Published  October 2019 Early access  April 2019

Fund Project: The author is supported by National Natural Science Foundation of China (No.11701290), Natural Science Youth Foundation of JiangSu Province, China(No.BK20170896), Natural Science Research Foundation of JiangSu Province, China(No. 17KJB110012) and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications, China(No. NY217150).

In this paper, a fully parabolic chemotaxis system for two species
$ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla w)+\mu_1u(1-u-a_1v),\ \ \ &x\in \Omega,\ t>0,\\ v_t = \Delta v-\chi_2\nabla\cdot(v\nabla w)+\mu_2v(1-v-a_2u),\ \ &x\in \Omega,\ t>0,\\ w_t = \Delta w-w+u+v,\ \ &x\in \Omega,\ t>0 \end{array}\right. \end{eqnarray*} $
is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain
$ \Omega\subset\mathbb{R}^n $
,
$ n\geq3 $
, with parameters
$ \chi_i, \mu_i, a_i>0 $
,
$ i = 1, 2 $
. It is shown that for some low energy initial data, the influence of chemotactic cross-diffusion coupled with proliferation may force some solutions to exceed any given threshold. Further, it is proved that if blow-up happens in a two-species chemotaxis(-growth) system, it is simultaneous for both of the chemotactic species.
Citation: Yan Li. Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5461-5480. doi: 10.3934/dcdsb.2019066
References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

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.

[3]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[4]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193.  doi: 10.1016/j.na.2012.04.035.

[5]

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.

[6]

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

[7]

E. E. Espejo ArenasA. Stevens and J. J. L. 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.

[8]

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.

[9]

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.

[10]

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

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[12]

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

[13]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564.

[14]

Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.  doi: 10.1016/j.na.2014.05.021.

[15]

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

[16]

I. G. PearceM. A. J. ChaplainP. G. SchofieldA. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.  doi: 10.1007/s00285-007-0088-4.

[17]

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

[18]

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.

[19]

X. Tang and Y. Tao, Analysis of a chemotaxis model for multi-species host-parasitoid interactions, Appl. Math. Sci., 2 (2008), 1239-1252. 

[20]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic keller-segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[21]

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.

[22]

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

[23]

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

[24]

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

[25]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[26]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[27]

M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[28]

M. Winkler, Finite-time blow-up in low-dimensional keller-segel systems with logistic-type superlinear degradation, Z. Angel. Math. Phy., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[29]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.

[30]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084.

[31]

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

show all references

References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

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.

[3]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[4]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193.  doi: 10.1016/j.na.2012.04.035.

[5]

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.

[6]

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

[7]

E. E. Espejo ArenasA. Stevens and J. J. L. 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.

[8]

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.

[9]

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.

[10]

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

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[12]

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

[13]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564.

[14]

Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.  doi: 10.1016/j.na.2014.05.021.

[15]

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

[16]

I. G. PearceM. A. J. ChaplainP. G. SchofieldA. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.  doi: 10.1007/s00285-007-0088-4.

[17]

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

[18]

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.

[19]

X. Tang and Y. Tao, Analysis of a chemotaxis model for multi-species host-parasitoid interactions, Appl. Math. Sci., 2 (2008), 1239-1252. 

[20]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic keller-segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[21]

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.

[22]

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

[23]

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

[24]

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

[25]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[26]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[27]

M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[28]

M. Winkler, Finite-time blow-up in low-dimensional keller-segel systems with logistic-type superlinear degradation, Z. Angel. Math. Phy., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[29]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.

[30]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084.

[31]

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

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