doi: 10.3934/dcdsb.2021218
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Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

2. 

School of Sciences, Southwest Petroleum University, Chengdu 610500, China

* Corresponding author: Yu Ma

Received  May 2021 Revised  July 2021 Early access September 2021

This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &x\in \Omega,\quad t>0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} $
in a bounded domain
$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $
with smooth boundary
$ \partial\Omega $
, where the parameters
$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $
are positive. It is shown that for any appropriate regular initial date
$ u_0 $
,
$ v_0 $
, the corresponding system possesses a global bounded classical solution in
$ n = 2 $
, and also in
$ n = 3 $
for
$ \chi $
being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if
$ b\lambda<\mu $
and the parameters
$ \chi $
and
$ \xi $
are sufficiently small, then the solution
$ (u,v,w) $
of this system converges to
$ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $
exponentially as
$ t\rightarrow \infty $
; if
$ b\lambda\geq \mu $
and
$ \chi $
is sufficiently small and
$ \xi $
is arbitrary, then the solution
$ (u,v,w) $
converges to
$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $
with exponential decay when
$ b\lambda> \mu $
, and with algebraic decay when
$ b\lambda = \mu $
.
Citation: Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021218
References:
[1]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

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T. Cie$\acute{s}$lak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[5]

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

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A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

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S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079.  Google Scholar

[8]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.   Google Scholar

[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.  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 chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.  Google Scholar

[13]

M. Herrero and J. Vel$\acute{a}$zquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[14]

E. Keller and L. 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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066.  Google Scholar

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X. Li and Y. Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.  doi: 10.1016/j.jmaa.2018.10.093.  Google Scholar

[19]

K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar

[20]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018.  Google Scholar

[21]

K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar

[22]

K. LinC. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007.  Google Scholar

[23]

G. LiY. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102.  Google Scholar

[24]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[25]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.  Google Scholar

[26]

M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278.  doi: 10.3934/dcdss.2020015.  Google Scholar

[27]

L. MiaoH. Yang and S. Fu, Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.  doi: 10.1016/j.aml.2020.106639.  Google Scholar

[28]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[29]

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

[30]

M. Negreanu and J. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar

[31]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[32]

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.  Google Scholar

[33]

C. StinnerJ. 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.  Google Scholar

[34]

S. QiuC. Mu and X. Tu, Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.  doi: 10.1016/j.nonrwa.2020.103190.  Google Scholar

[35]

X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp. doi: 10.1007/s00033-020-01413-6.  Google Scholar

[36]

X. TuC. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar

[37]

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

[38]

J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[39]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002.  Google Scholar

[40]

Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.  Google Scholar

[41]

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.  Google Scholar

[42]

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

[43]

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

[44]

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.  Google Scholar

[45]

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.  Google Scholar

[46]

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.  Google Scholar

[47]

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.  Google Scholar

[48]

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

[49]

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

[50]

M. Winkler, How far can chemotaxis 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]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

T. Cie$\acute{s}$lak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[5]

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

[6]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

[7]

S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079.  Google Scholar

[8]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.   Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

M. HirataS. KurimaM. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.  Google Scholar

[13]

M. Herrero and J. Vel$\acute{a}$zquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[14]

E. Keller and L. 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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066.  Google Scholar

[18]

X. Li and Y. Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.  doi: 10.1016/j.jmaa.2018.10.093.  Google Scholar

[19]

K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar

[20]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018.  Google Scholar

[21]

K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar

[22]

K. LinC. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007.  Google Scholar

[23]

G. LiY. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102.  Google Scholar

[24]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[25]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.  Google Scholar

[26]

M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278.  doi: 10.3934/dcdss.2020015.  Google Scholar

[27]

L. MiaoH. Yang and S. Fu, Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.  doi: 10.1016/j.aml.2020.106639.  Google Scholar

[28]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[29]

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

[30]

M. Negreanu and J. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar

[31]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[32]

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.  Google Scholar

[33]

C. StinnerJ. 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.  Google Scholar

[34]

S. QiuC. Mu and X. Tu, Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.  doi: 10.1016/j.nonrwa.2020.103190.  Google Scholar

[35]

X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp. doi: 10.1007/s00033-020-01413-6.  Google Scholar

[36]

X. TuC. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar

[37]

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

[38]

J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[39]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002.  Google Scholar

[40]

Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.  Google Scholar

[41]

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.  Google Scholar

[42]

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

[43]

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

[44]

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.  Google Scholar

[45]

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.  Google Scholar

[46]

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.  Google Scholar

[47]

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.  Google Scholar

[48]

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

[49]

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

[50]

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