# American Institute of Mathematical Sciences

February  2020, 13(2): 293-319. doi: 10.3934/dcdss.2020017

## Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

 Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849, USA

* Corresponding author: Rachidi B. Salako

Received  February 2017 Revised  April 2018 Published  January 2019

The current paper is devoted to the study of traveling wave solutions of the following parabolic-elliptic-elliptic chemotaxis systems,
 $$$\label{main-eq-abstract}\begin{cases}u_{t} = Δ u- \nabla · ({ χ_1 u} \nabla v_1)+ \nabla · ({ χ_2 u} \nabla v_2) + u(a-bu), \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_1-λ_1v_1+μ_1u, \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_2-λ_2v_2+μ_2u, \;\;\;\; x∈\mathbb{R}^N, \end{cases}\;\;\;\;\;\;\;\;(0.1)$$$
where
 $a>0, \ b>0,$
 $u(x, t)$
represents the population density of a mobile species,
 $v_1(x, t),$
represents the population density of a chemoattractant,
 $v_2(x, t)$
represents the population density of a chemorepulsion, the constants
 $χ_1≥ 0$
and
 $χ_2≥ 0$
represent the chemotaxis sensitivities, and the positive constants
 $λ_1, λ_2, μ_1$
, and
 $μ_2$
are related to growth rate of the chemical substances. It was proved in an earlier work by the authors of the current paper that there is a nonnegative constant
 $K$
depending on the parameters
 $χ_1, μ_1, λ_1, χ_2, μ_2$
, and
 $λ_2$
such that if
 $b+χ_2μ_2>χ_1μ_1+K$
, then the positive constant steady solution
 $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$
of (0.1) is asymptotically stable with respect to positive perturbations. In the current paper, we prove that if
 $b+χ_2μ_2>χ_1μ_1+K$
, then there exists a positive number
 $c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)≥ 2\sqrt{a}$
such that for every
 $c∈ ( c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)\ , \ ∞)$
and
 $ξ∈ S^{N-1}$
, the system has a traveling wave solution
 $(u(x, t), v_1(x, t), v_2(x, t)) = (U(x·ξ-ct), V_1(x·ξ-ct), V_2(x·ξ-ct))$
with speed
 $c$
connecting the constant solutions
 $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$
and
 $(0, 0, 0)$
, and it does not have such traveling wave solutions of speed less than
 2\sqrt a $. Moreover we prove that $\begin{equation*}\lim\limits_{(χ_{1}, χ_2)?(0^+, 0^+)}c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2) = \begin{cases}\ 2\sqrt{a} \;\;\text{if}\;\; a≤ \min\{λ_1, λ_2\}\\\frac{a+λ_1}{\sqrt{λ_1}} \;\;\text{if}\;\; λ_1≤ \min\{a, λ_2\}\\\frac{a+λ_2}{\sqrt{λ_2}} \;\;\text{if}\;\; λ_2≤ \min\{a, λ_1\}\end{cases}\end{equation*}$for every $ λ_1, λ_2, μ_1, μ_2>0$, and $\begin{equation*}\lim\limits_{x?∞}\frac{U(x)}{e^{-\sqrt a μ x}} = 1, \end{equation*}$where $μ$is the only solution of the equation $μ+\frac{1}{μ} = \frac{c}{\sqrt{a}}$in the interval $(0\ , \ \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$. Citation: Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 293-319. doi: 10.3934/dcdss.2020017 ##### References:  [1] S. Ai, W. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21. doi: 10.3934/dcdsb.2015.20.1. [2] S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737. doi: 10.3934/mbe.2015.12.717. [3] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [4] 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. [5] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26. [6] 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. [7] H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, 2015. . [8] M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285. [9] T. Cieślak, P. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Parabolic and Navier-Stokes Equations Banach Center Publications, Institute of Mathematics Polish Academy of Sciences Warszawa, 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [10] 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. [11] 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. [12] E. Espejoand and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [13] 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. [14] 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. [15] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. [16] A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [17] M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxisgrowth model, Interfaces Free Bound, 8 (2006), 223-245. doi: 10.4171/IFB/141. [18] 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. [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. [20] M. A. Herrero and J. J. L. Velasquez, A blow-up mechanism for a chemotaxis model, Annali Della Scuola Normale Superiore di Pisa, Classe di Scienze, 24 (1997), 633-683. [21] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlin. Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. [22] D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25. doi: 10.1007/s00332-003-0548-y. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] J. Li, T. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849. doi: 10.1142/S0218202514500389. [29] 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. [30] 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. [31] K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attractionrepulsion chemotaxis with non-linear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [32] 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. [33] P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [34] 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. [35] B. P. Marchant, J. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxisdominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671. doi: 10.1088/0951-7715/14/6/313. [36] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612. doi: 10.1137/0505061. [37] 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. [38] G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538. doi: 10.4171/IFB/200. [39] 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. [40] 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. [41] 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. [42] R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on{{\mathbb{R}}^{N}}$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237–2273, https://arXiv.org/pdf/1709.05785.pdf. doi: 10.1142/S0218202518400146. [43] 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), https://doi.org/10.1007/s10884-017-9602-6. [44] R. B. Salako and W. Shen, Existence of Traveling wave solution of the full parabolic chemotaxis system, Nonlinear Analysis: Real World Applications, 42 (2018), 93-119. [45] 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. [46] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on${{\mathbb{R}}^{N}}$, Journal of Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011. [47] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0. [48] 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. [49] W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. [50] 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. [51] 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. [52] 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. [53] 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. [54] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. doi: 10.1215/kjm/1250522506. [55] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. [56] Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024. [57] Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attractionrepulsion chemotaxis system. Discrete Contin, Dyn. Syst. Ser. B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031. [58] Z. A. Wang, Mathematics of traveling waves in chemotaxis–review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. [59] H. F. Weinberger, Long–time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [60] 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. [61] 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. [62] 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. [63] 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. [64] 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. [65] 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. [66] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 1125-1133. doi: 10.3934/proc.2015.1125. [67] Q. Zhang and Y. Li, An attraction–repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311. [68] P. Zheng, C. Mu and X. Hu, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202. doi: 10.1016/j.camwa.2016.08.028. [69] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031. [70] A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl.(9), 98 (2012), 89-102. doi: 10.1016/j.matpur.2011.11.007. show all references ##### References:  [1] S. Ai, W. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21. doi: 10.3934/dcdsb.2015.20.1. [2] S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737. doi: 10.3934/mbe.2015.12.717. [3] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [4] 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. [5] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26. [6] 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. [7] H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, 2015. . [8] M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285. [9] T. Cieślak, P. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Parabolic and Navier-Stokes Equations Banach Center Publications, Institute of Mathematics Polish Academy of Sciences Warszawa, 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [10] 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. [11] 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. [12] E. Espejoand and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [13] 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. [14] 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. [15] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. [16] A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [17] M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxisgrowth model, Interfaces Free Bound, 8 (2006), 223-245. doi: 10.4171/IFB/141. [18] 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. [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. [20] M. A. Herrero and J. J. L. Velasquez, A blow-up mechanism for a chemotaxis model, Annali Della Scuola Normale Superiore di Pisa, Classe di Scienze, 24 (1997), 633-683. [21] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlin. Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. [22] D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25. doi: 10.1007/s00332-003-0548-y. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] J. Li, T. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849. doi: 10.1142/S0218202514500389. [29] 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. [30] 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. [31] K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attractionrepulsion chemotaxis with non-linear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [32] 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. [33] P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [34] 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. [35] B. P. Marchant, J. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxisdominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671. doi: 10.1088/0951-7715/14/6/313. [36] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612. doi: 10.1137/0505061. [37] 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. [38] G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538. doi: 10.4171/IFB/200. [39] 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. [40] 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. [41] 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. [42] R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on{{\mathbb{R}}^{N}}$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237–2273, https://arXiv.org/pdf/1709.05785.pdf. doi: 10.1142/S0218202518400146. [43] 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), https://doi.org/10.1007/s10884-017-9602-6. [44] R. B. Salako and W. Shen, Existence of Traveling wave solution of the full parabolic chemotaxis system, Nonlinear Analysis: Real World Applications, 42 (2018), 93-119. [45] 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. [46] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on${{\mathbb{R}}^{N}}$, Journal of Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011. [47] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0. [48] 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. [49] W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. [50] 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. [51] 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. [52] 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. [53] 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. [54] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. doi: 10.1215/kjm/1250522506. [55] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. [56] Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024. [57] Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attractionrepulsion chemotaxis system. Discrete Contin, Dyn. Syst. Ser. B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031. [58] Z. A. Wang, Mathematics of traveling waves in chemotaxis–review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. [59] H. F. Weinberger, Long–time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [60] 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. [61] 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. [62] 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. [63] 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. [64] 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. [65] 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. [66] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 1125-1133. doi: 10.3934/proc.2015.1125. [67] Q. Zhang and Y. Li, An attraction–repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311. [68] P. Zheng, C. Mu and X. Hu, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202. doi: 10.1016/j.camwa.2016.08.028. [69] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031. [70] A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. 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