Korean J. Math.  Vol 24, No 2 (2016)  pp.215-233
DOI: https://doi.org/10.11568/kjm.2016.24.2.215

Interactive dynamics in a bistable attraction-repulsion chemotaxis system

YoonMee Ham, Sang-Gu Lee


We consider a bistable attraction-repulsion chemotaxis system in one dimension. The study in this paper asserts that conditions for chemotactic coefficients for attraction and repulsion to show existence of stationary solutions and Hopf bifurcation in the interfacial problem as the bifurcation parameters vary are obtained analytically.


ttraction-repulsion, chemotaxis, free boundary problem, Hopf bifurcation.

Subject classification

35K57, 35R35, 35B32, 35B25, 35K55, 35K57, 58J55.


Full Text:



M.Aida, T.Tsujikawa, M.Efendiev, A.Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. 74 (2006), 453–474. (Google Scholar)

M.G.Crandall and P.H.Rabinowitz, The Hopf Bifurcation Theorem in Infinite Dimensions, Arch. Rat. Mech. Anal. 67 (1978), 53–72. (Google Scholar)

P.Fife, Dynamics of internal layers and diffusive interfaces, CMBS-NSF Regional Conference Series in Applied Mathematics, 53, Philadelphia: SIAM 1988. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

H. Jin and Z.A. Wang, Boundedness, blowup and critical mass phe- nomenon in competing chemotaxis, J. Differential Equations (2015), http://dx.doi.org/10.1016/j.jde.2015.08.040 (Google Scholar)

S.Kawaguchi, Chemotaxis-growth under the influence of lateral inhibition in a three-component reaction-diffusion system, Nonlinearity, 24 (2011), 1011–1031. (Google Scholar)

J.P.Keener, A geometrical theory for spiral waves in excitable media, SIAM J. Appl. Math. 46 (1986), 1039–1056. (Google Scholar)

E.F.Keller and L.A.Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415. (Google Scholar)

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

M.Luca, A.Chavez-Ross, L.Edelstein-Keshet and A.Mogilner, Chemotactic sig- nalling, microglia, and alzheimer’s disease senile plague: is there a connection?, Bull. Math. Biol. 65 (2003), 215–225. (Google Scholar)

H. P. McKean, Nagumo’s equation, Adv. in Math. 4 (1975), 209–223. (Google Scholar)

M.Mimura and T.Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A. 230 (1996), 499–543. (Google Scholar)

H.Mori, Global existence of the Cauchy problem for a chemotactic system with prey-predator dynamics, Hiroshima Math. J. 36 (2006), 77–111. (Google Scholar)

K.Painter an d T.Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart. 10 (2002), 501–543. (Google Scholar)

Y.Tao and Z.Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci. 23, (2013), DOI: 10.1142/S0218202512500443 (Google Scholar)

T.Tsujikawa, Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth, Methods Appl. Anal. 3 (1996), 401–431. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr