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October  2017, 10(5): 1165-1174. doi: 10.3934/dcdss.2017063

Stability and bifurcation analysis in a chemotaxis bistable growth system

Y. Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China

* Corresponding author: Ping Liu

Received  September 2016 Revised  January 2017 Published  June 2017

Fund Project: Partially supported by NSFC grant 11571086,11471091 and Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province LC2013C01

The stability analysis of a chemotaxis model with a bistable growth term in both unbounded and bounded domains is studied analytically. By the global bifurcation theorem, we identify the full parameter regimes in which the steady state bifurcation occurs.

Citation: Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063
References:
[1]

J. Adler, Chemotaxis in bacteria, Science (New York, NY), 153 (1966), 708-716. Google Scholar

[2]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, Journal of the London Mathematical Society, 74 (2006), 453-474. doi: 10.1112/S0024610706023015. Google Scholar

[3]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. Google Scholar

[4]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149. Google Scholar

[5]

A. GambaD. AmbrosiA. Conigliov De CandiaS. Di TaliaE. GiraudoG. SeriniL. Preziosi and F. Bussolino, Percolation, morphogenesis, and burgers dynamics in blood vessels formation, Physical Review Letters, 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[6]

R. E. Goldstein, Traveling-wave chemotaxis, Physical Review Letters, 77 (1996), 775-778. doi: 10.1103/PhysRevLett.77.775. Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[8]

D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165. Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[10]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[11]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A: Statistical Mechanics and its Applications, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9. Google Scholar

[12]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. Google Scholar

[13]

M. NelkinS. MeneveauK. R. Sreenivasan and C. K. Peng, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49. Google Scholar

[14]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar

[15]

K. J. PainterP. K. Maini and H. G. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized turing mechanism with chemotaxis, Proceedings of the National Academy of Sciences, 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549. Google Scholar

[16]

K. J. PainterP. K. Maini and H. G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bulletin of Mathematical Biology, 62 (2000), 501-525. Google Scholar

[17]

C. S. Patlak, Random walk with persistence and external bias, Bulletin of Mathematical Biology, 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[18]

G. J. PettetH. M. ByrneD. L. S. McElwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue, Mathematical Biosciences, 136 (1996), 35-63. doi: 10.1016/0025-5564(96)00044-2. Google Scholar

[19]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[20]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proceedings of the National Academy of Sciences, 98 (2001), 14907-14912. doi: 10.1073/pnas.261574598. Google Scholar

[21]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science (New York, NY), 153 (1966), 708-716. Google Scholar

[2]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, Journal of the London Mathematical Society, 74 (2006), 453-474. doi: 10.1112/S0024610706023015. Google Scholar

[3]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. Google Scholar

[4]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149. Google Scholar

[5]

A. GambaD. AmbrosiA. Conigliov De CandiaS. Di TaliaE. GiraudoG. SeriniL. Preziosi and F. Bussolino, Percolation, morphogenesis, and burgers dynamics in blood vessels formation, Physical Review Letters, 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[6]

R. E. Goldstein, Traveling-wave chemotaxis, Physical Review Letters, 77 (1996), 775-778. doi: 10.1103/PhysRevLett.77.775. Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[8]

D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165. Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[10]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[11]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A: Statistical Mechanics and its Applications, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9. Google Scholar

[12]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002. Google Scholar

[13]

M. NelkinS. MeneveauK. R. Sreenivasan and C. K. Peng, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49. Google Scholar

[14]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar

[15]

K. J. PainterP. K. Maini and H. G. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized turing mechanism with chemotaxis, Proceedings of the National Academy of Sciences, 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549. Google Scholar

[16]

K. J. PainterP. K. Maini and H. G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bulletin of Mathematical Biology, 62 (2000), 501-525. Google Scholar

[17]

C. S. Patlak, Random walk with persistence and external bias, Bulletin of Mathematical Biology, 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[18]

G. J. PettetH. M. ByrneD. L. S. McElwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue, Mathematical Biosciences, 136 (1996), 35-63. doi: 10.1016/0025-5564(96)00044-2. Google Scholar

[19]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[20]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proceedings of the National Academy of Sciences, 98 (2001), 14907-14912. doi: 10.1073/pnas.261574598. Google Scholar

[21]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. Google Scholar

Figure 1.  Basic phase portrait of (2)
Figure 2.  Parameter space for Turing instability. The parameter values are $d_1=d_2=f=g=1, \nu=\frac{1}{4}, $$ M=\frac{(\sqrt{d_1g}+\sqrt{(1-\nu)d_2})^2}{f}$
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