Acknowledgement
This work was supported by Kyonggi University Research Grant 2021.
References
- M.G. Crandall and P.H. Rabinowitz, The Hopf Bifurcation Theorem in Infinite Dimensions, Arch. Rat. Mech. Anal. 67 (1978), 53-72. https://doi.org/10.1007/BF00280827
- P. Fife, Dynamics of internal layers and diffusive interfaces, CMBS-NSF Regional Conference Series in Applied Mathematics 53, Philadelphia: SIAM, 1988.
- K. Ikeda and M. Mimura, Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion, Commun. Pur. Appl. Anal. 11 (2012), 275-305. https://doi.org/10.3934/cpaa.2012.11.275
- Y.M. Ham-Lee, R. Schaaf and R. Thompson, A Hopf bifurcation in a parabolic free boundary problem, J. of Comput. Appl. Math. 52 (1994), 305-324. https://doi.org/10.1016/0377-0427(94)90363-8
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics No. 840 (Springer-Verlag, New York Heidelberg Berlin, 1981).
- 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. https://doi.org/10.1007/s00332-010-9082-x
- H.Y. Jin, J. Li, Z.A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations 55 (2013), 193-219. https://doi.org/10.1016/j.jde.2013.04.002
- S. Kwaguchi and M. Mimura, Collision of traveling waves in a reaction-diffusion system with global coupling effect, SIAM J. Appl. Math. 59 (1999), 920-941. https://doi.org/10.1137/S003613999630664X
- S. Kawaguchi, Chemotaxis-growth under the influence of lateral inhibition in a three-component reaction-diffusion system, Nonlinearity 24 (2011), 1011-1031. https://doi.org/10.1088/0951-7715/24/4/002
- M.Luca, A.Chavez-Ross, L.Edelstein-Keshet, A.Mogilner,Chemotactic signalling, microglia, and alzheimer's disease senile plague: is there a connection ?, Bull. Math. Biol. 65 (2003), 215-225. https://doi.org/10.1016/S0092-8240(03)00030-2
- H.P. McKean, Nagumo's equation, Adv. Math. 4 (1975), 209-223. https://doi.org/10.1016/0001-8708(70)90023-X
- M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A 230 (1996), 499-543. https://doi.org/10.1016/0378-4371(96)00051-9
- U. Middya, M.D. Graham, D. Luss, and M. Sheintuch, Pattern selection in controlled reation-diffusion systems, J. Chem. Phys. 98 (1993), pp. 2823-2836. https://doi.org/10.1063/1.464111
- K.Painter, T.Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart. 10 (2002), 501-543.
- R. Schaaf, Statioltary sobltions of chemotaxis equatiotts, Trans. AMS 292 (1985), 531-556. https://doi.org/10.1090/S0002-9947-1985-0808736-1
- Y. Tao and M. Winkler Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals, Discrete & Continuous Dynamical Systems-B 20 (2015), 3165-3183. https://doi.org/10.3934/dcdsb.2015.20.3165
- Y. Tao, Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci. 23 (2013), DOI: 10.1142/S0218202512500443
- T. Tsujikawa, Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth, Methods Appl. Anal. 3 (1996), 401-431. https://doi.org/10.4310/MAA.1996.v3.n4.a1
- G. veser, F. Mertens, A. S. Mikhailov, and R. Imbihl, Global coupling in the presence of defects : Synchronization in an oscillatory surface reation, Phys. Rev. Lett. 77 (1993), 935-938. https://doi.org/10.1103/PhysRevLett.71.935
- Y. Zhu and F. Cong, Global existence to an attraction-repulsion chemotaxis model with fast diffusion and nonlinear source, Discrete Dynamics in nature and society, 2015, Article ID 143718, 8pages.