Abstract
Chemotaxis is a type of oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider local existence and stability of nontrivial steady states of a logistic type of chemotaxis. We carry out the bifurcation theory to obtain the local existence of the steady state and apply the expansion method on the chemotaxis to investigate the bifurcation direction. Moreover, by applying the bifurcation direction, we obtain the bifurcating steady state is stable when the bifurcation curve turns to right under certain conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chertock, A., Kurganov, A., Wang, X., Wu, Y. On a chemotaxis model with saturated chemotactic flux. Kinetic and Related Models, 3: 51–95 (2012)
Crandall, M., Rabinowitz, P. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal., 52: 161–181 (1973)
Crandall, M.G., Rabinowitz, P. Bifurcation from simple eigenvalues. J. Functional Analysis, 8: 321–340 (1971)
Hillen, T., Painter, K.J. A user’s guide to PDE models for chemotaxis. J. Math. Biol., 58: 183–217 (2009)
Horstmann, D. From 1970 until now: The Keller-Segal model in chemotaxis and its consequences I and II. Jahresber. DMV, 105: 103–165 (2003); 106: 51–69 (2004)
Keller, E., Segel, L. Initiation of slime mold aggregation viewed as an instability. J. Theoret Biol., 26: 399–415 (1970)
Kuto, K., Osaki, K., Sakurai, T., Tsujikawa, T. Spatial pattern formation in a chemotaxis-diffusion-growth model. Physica D., 241: 1629–1639 (2012)
Mimura, M., Tsujikawa, T. Aggregating pattern dynamics in a chemotaxis model including growth. Physica A., 230: 499–543 (1996)
Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M. Exponential attracktor for a chemotaxis-growth system of equations. Nonlinear Analysis: Theory Methods and Applications, 51: 119–144 (2002)
Painter, K.J., Hillen, T. Spatio-temporal chaos in a chemotaxis model. Physica D., 240: 363–375 (2011)
Shi, J., Wang, X. On the global bifurcation for quasilinear elliptic systems on bounded domains. J. Diff. Eqs., 246: 2788–2812 (2009)
Tello, J.I., Winkle, M. A Chemotaxis system with logistic source. Communications in Partial Differential Equations, 32: 849–877 (2007)
Wang, X., Xu, Q. Spiky and transition layer steady states of chemotaxis systems via global bifurcation and helly’s compactness theorem. J. Math. Biol., 66: 1241–1266 (2013)
Acknowledgements
The authors are greatly indebted to Prof. Xuefeng Wang and Prof. Yaping Wu for their encouragement, valuable suggestions and helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the National Natural Science Foundation of China (No.11501031,11471221,71373023), Beijing Natural Science Foundation (1132003 and KZ201310028030) and Zhejiang A&F University telant program (2013FR078).
Rights and permissions
About this article
Cite this article
Cai, Cq., Xu, Q. & Liu, Xl. The local bifurcation and stability of nontrivial steady states of a logistic type of chemotaxis. Acta Math. Appl. Sin. Engl. Ser. 33, 799–808 (2017). https://doi.org/10.1007/s10255-016-0500-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-016-0500-1