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October  2019, 39(10): 5945-5973. doi: 10.3934/dcds.2019260

Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

Rachidi B. Salako ,

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: Rachidi B. Salako (salako.7@osu.edu)

Received  December 2018 Published  July 2019

In this paper, we study traveling wave solutions of the chemotaxis system
$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad \ x\in \mathbb{R} \\ \tau {{\partial }_{t}}{{v}_{1}} = (\Delta -{{\lambda }_{1}}I){{v}_{1}}+{{\mu }_{1}}u,\qquad \ x\in \mathbb{R}, \\ \tau \partial {{v}_{2}} = (\Delta -{{\lambda }_{2}}I){{v}_{2}}+{{\mu }_{2}}u,\qquad \ \ x\in \mathbb{R}, \\\end{array} \right. & (0.1) \\\end{matrix} $
where
$ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $
 (
$ i = 1,2 $
) and
$ \ a>0,\ b> 0 $
 are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant
$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) $
 such that for every
$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2, \lambda_2) $
, (0.1) has a traveling wave solution
$ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $
 connecting
$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $
 and
$ (0,0,0) $
 satisfying
$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $
where
$ \mu\in (0,\sqrt a) $
 is such that
$ c = c_\mu: = \mu+\frac{a}{\mu} $
. Moreover,
$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $
and
$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $
where
$ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $
. We also show that (1) has no traveling wave solution connecting
$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $
 and
$ (0,0,0) $
 with speed
$ c<2\sqrt{a} $
.
Keywords: Parabolic-parabolic-parabolic chemotaxis system, logistic source, classical solution, local existence, global existence, asymptotic stability, traveling wave solutions.
Mathematics Subject Classification: 35B35, 35B40, 35K57, 35Q92, 92C17.
Citation: Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260
References:
[1]

N. Bellomo, A. Bellouquid, Y. 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

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H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excita media, Journal of Functional Analysis, 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

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H. Berestycki, F. Hamel and and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅰ - Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.  Google Scholar

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H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ - General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.  Google Scholar

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H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. Google Scholar

[6]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680.   Google Scholar

[7]

J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization Techniques on Unbounded Domains: Application to a Chemotaxis System on $\mathbb{R}^{N}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.  Google Scholar

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R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

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M. Freidlin, On wave front propagation in periodic media. In: Stochastic Analysis and Applications, ed. M. Pinsky, Advances in Probablity and Related Topics, 7 (1984), 147-166.  Google Scholar

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M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

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A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

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E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.  Google Scholar

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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

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

[16]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.  Google Scholar

[17]

D. Horstmann, From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.   Google Scholar

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H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

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K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

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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.  Google Scholar

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E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

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A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[23]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[24]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[25]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.  Google Scholar

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M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimers disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar

[27]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[28]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser Inequality to a Parabolic System of Chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.   Google Scholar

[29]

J. Nolen, M. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[30]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.  Google Scholar

[31]

R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$, Journal of Dynamics and Differential Equations, (2017). doi: 10.1007/s10884-017-9602-6.  Google Scholar

[32]

R. B. Salako and W. Shen, Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source, Discrete and Continuous Dynamical System-Series S. doi: 10.3934/dcdss.2020017.  Google Scholar

[33]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete and Continuous Dynamical Systems - Series A, 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.  Google Scholar

[34]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.  Google Scholar

[35]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[36]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.   Google Scholar

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[38]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

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

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Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Dyn. Syst. Ser. B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

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[43]

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

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[45]

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

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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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show all references

References:
[1]

N. Bellomo, A. Bellouquid, Y. 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

[2]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excita media, Journal of Functional Analysis, 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[3]

H. Berestycki, F. Hamel and and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅰ - Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.  Google Scholar

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ - General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.  Google Scholar

[5]

H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. Google Scholar

[6]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680.   Google Scholar

[7]

J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization Techniques on Unbounded Domains: Application to a Chemotaxis System on $\mathbb{R}^{N}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.  Google Scholar

[8]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[9]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic Analysis and Applications, ed. M. Pinsky, Advances in Probablity and Related Topics, 7 (1984), 147-166.  Google Scholar

[10]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

[11]

A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[12]

E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.  Google Scholar

[14]

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

[15]

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

[16]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.  Google Scholar

[17]

D. Horstmann, From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.   Google Scholar

[18]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[19]

K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

[20]

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.  Google Scholar

[21]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[22]

A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[23]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[24]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[25]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.  Google Scholar

[26]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimers disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar

[27]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[28]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser Inequality to a Parabolic System of Chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.   Google Scholar

[29]

J. Nolen, M. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[30]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.  Google Scholar

[31]

R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$, Journal of Dynamics and Differential Equations, (2017). doi: 10.1007/s10884-017-9602-6.  Google Scholar

[32]

R. B. Salako and W. Shen, Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source, Discrete and Continuous Dynamical System-Series S. doi: 10.3934/dcdss.2020017.  Google Scholar

[33]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete and Continuous Dynamical Systems - Series A, 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.  Google Scholar

[34]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.  Google Scholar

[35]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[36]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.   Google Scholar

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[38]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[39]

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

[40]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Dyn. Syst. Ser. B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

[41]

H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[42]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[43]

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

[44]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[45]

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

[46]

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

[47]

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