November  2019, 24(11): 6005-6024. doi: 10.3934/dcdsb.2019118

Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

* Corresponding author: niu@hit.edu.cn (Ben Niu)

Received  November 2018 Revised  December 2018 Published  June 2019

Fund Project: This research is supported by National Natural Science Foundation of China (11701120, 11771109) and the Foundation for Innovation at HIT(WH)

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

Citation: Yuxiao Guo, Ben Niu. Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6005-6024. doi: 10.3934/dcdsb.2019118
References:
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Q. An and W. Jiang, Hopf-zero bifurcation and the normal forms in reaction-diffusion systems with time delays, preprint, arXiv: 1710.10411. Google Scholar

[2]

S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, International Journal of Bifurcation and Chaos, 22 (2012), 1250061, 11pp. doi: 10.1142/S0218127412500617.  Google Scholar

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J. D. FerreiraA. P. G. Nieva and W. M. Yepez, Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations, Journal of Mathematical Analysis and Applications, 455 (2017), 1-51.  doi: 10.1016/j.jmaa.2017.05.040.  Google Scholar

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B. Niu and W. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, Journal of Mathematical Analysis and Applications, 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.  Google Scholar

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L. A. Segel and J. L. Jackson, Dissipative structure: an explanation and an ecological example, Journal of Theoretical Biology, 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.  Google Scholar

[23]

Z. Song and J. Xu, Bursting near Bautin bifurcation in a neural network with delay coupling., International Journal of Neural Systems, 19 (2009), 359-373.  doi: 10.1142/S0129065709002087.  Google Scholar

[24]

Y. SongT. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 229-258.  doi: 10.1016/j.cnsns.2015.10.002.  Google Scholar

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[26]

G. Sun, Z. Jin and Q. Liu, et al., Spatial pattern in an epidemic system with cross-diffusion of the susceptible, Journal of Biological Systems, 17 (2009), 141–152. doi: 10.1142/S0218339009002843.  Google Scholar

[27]

J. Wang and Y. Wang, Bifurcation analysis in a diffusive Segel-Jackson model, Journal of Mathematical Analysis and Applications, 415 (2014), 204-216.  doi: 10.1016/j.jmaa.2014.01.070.  Google Scholar

[28]

X. Wei and J. Wei, Turing instability and bifurcation analysis in a diffusive bimolecular system with delayed feedback, Communications in Nonlinear Science and Numerical Simulation, 50 (2017), 241-255.  doi: 10.1016/j.cnsns.2017.03.006.  Google Scholar

[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[30]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007%2F978-1-4612-4050-1.  Google Scholar

[31]

X. YangM. YangH. Liu and et al., Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms, Chaos, Solitons & Fractals, 38 (2008), 575-589.  doi: 10.1016/j.chaos.2007.01.001.  Google Scholar

[32]

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

[33]

B. Zhen and J. Xu, Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 442-458.  doi: 10.1016/j.cnsns.2009.04.006.  Google Scholar

show all references

References:
[1]

Q. An and W. Jiang, Hopf-zero bifurcation and the normal forms in reaction-diffusion systems with time delays, preprint, arXiv: 1710.10411. Google Scholar

[2]

S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, International Journal of Bifurcation and Chaos, 22 (2012), 1250061, 11pp. doi: 10.1142/S0218127412500617.  Google Scholar

[3]

Y. Du, B. Niu, Y. Guo and J. Wei, Double Hopf bifurcation in delayed reaction-diffusion systems, Journal of Dynamics and Differential Equation, 2019, 1–46. doi: 10.1007/s10884-018-9725-4.  Google Scholar

[4]

B. Ermentrout and J. D. Drover, Nonlinear coupling near a degenerate Hopf (Bautin) bifurcation, SIAM Journal on Applied Mathematics, 63 (2003), 1627-1647.  doi: 10.1137/S0036139902412617.  Google Scholar

[5]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications, 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.  Google Scholar

[6]

J. D. FerreiraA. P. G. Nieva and W. M. Yepez, Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations, Journal of Mathematical Analysis and Applications, 455 (2017), 1-51.  doi: 10.1016/j.jmaa.2017.05.040.  Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[8]

S. GuoY. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, Journal of Differential Equations, 244 (2008), 444-486.  doi: 10.1016/j.jde.2007.09.008.  Google Scholar

[9]

W. Jiang, H. Wang and X. Cao, Turing instability and turing-hopf bifurcation in diffusive schnakenberg systems with gene expression time delay, Journal of Dynamics and Differential Equations, 2018, 1–25. doi: 10.1007/s10884-018-9702-y.  Google Scholar

[10]

W. Jiang and Y. Yuan, Bogdanov-Takens singularity in Van der Pol's oscillator with delayed feedback, Physica D: Nonlinear Phenomena, 227 (2007), 149-161.  doi: 10.1016/j.physd.2007.01.003.  Google Scholar

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007%2F978-1-4612-4342-7.  Google Scholar

[12] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981.   Google Scholar
[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[14]

A. V. Ion, On the Bautin bifurcation for systems of delay differential equations, Acta Univ. Apulensis, 8 (2004), 235-246.   Google Scholar

[15]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004. doi: 10.1007%2F978-1-4757-3978-7.  Google Scholar

[16]

X. LinJ. So and J. Wu, Centre manifolds for partial differential equations with delays, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 122 (1992), 237-254.  doi: 10.1017/S0308210500021090.  Google Scholar

[17]

X. Liu and S. Liu, Codimension-two bifurcation analysis in two-dimensional Hindmarsh-Rose model, Nonlinear Dynamics, 67 (2012), 847-857.  doi: 10.1007/s11071-011-0030-6.  Google Scholar

[18]

Z. Lü and L. Duan, Codimension-2 Bautin bifurcation in the Lü system, Physics Letters A, 366 (2007), 442-446.  doi: 10.1016/j.physleta.2007.02.047.  Google Scholar

[19]

B. Niu and W. Jiang, Multiple bifurcation analysis in a NDDE arising from van der Pol's equation with extended delay feedback, Nonlinear Analysis: Real World Applications, 14 (2013), 699-717.  doi: 10.1016/j.nonrwa.2012.07.028.  Google Scholar

[20]

B. Niu and W. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, Journal of Mathematical Analysis and Applications, 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.  Google Scholar

[21]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A, 10 (2003), 863-874.   Google Scholar

[22]

L. A. Segel and J. L. Jackson, Dissipative structure: an explanation and an ecological example, Journal of Theoretical Biology, 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.  Google Scholar

[23]

Z. Song and J. Xu, Bursting near Bautin bifurcation in a neural network with delay coupling., International Journal of Neural Systems, 19 (2009), 359-373.  doi: 10.1142/S0129065709002087.  Google Scholar

[24]

Y. SongT. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 229-258.  doi: 10.1016/j.cnsns.2015.10.002.  Google Scholar

[25]

Y. SuJ. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, Journal of Differential Equations, 247 (2009), 1156-1184.  doi: 10.1016/j.jde.2009.04.017.  Google Scholar

[26]

G. Sun, Z. Jin and Q. Liu, et al., Spatial pattern in an epidemic system with cross-diffusion of the susceptible, Journal of Biological Systems, 17 (2009), 141–152. doi: 10.1142/S0218339009002843.  Google Scholar

[27]

J. Wang and Y. Wang, Bifurcation analysis in a diffusive Segel-Jackson model, Journal of Mathematical Analysis and Applications, 415 (2014), 204-216.  doi: 10.1016/j.jmaa.2014.01.070.  Google Scholar

[28]

X. Wei and J. Wei, Turing instability and bifurcation analysis in a diffusive bimolecular system with delayed feedback, Communications in Nonlinear Science and Numerical Simulation, 50 (2017), 241-255.  doi: 10.1016/j.cnsns.2017.03.006.  Google Scholar

[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[30]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007%2F978-1-4612-4050-1.  Google Scholar

[31]

X. YangM. YangH. Liu and et al., Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms, Chaos, Solitons & Fractals, 38 (2008), 575-589.  doi: 10.1016/j.chaos.2007.01.001.  Google Scholar

[32]

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

[33]

B. Zhen and J. Xu, Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 442-458.  doi: 10.1016/j.cnsns.2009.04.006.  Google Scholar

Figure 1.  The Bautin bifurcation diagram near the equilibrium for a) $ l_2(0)<0 $ and b) $ l_2(0)>0 $
Figure 2.  $ a = 5 $. a) the first Lyapunov coefficient $ l_1 $ varies with respect to $ k $. b) the Bautin bifurcation diagram near the Bautin point $ (k^\ast, \tau^\ast) = (0.3075, 0.6543) $
Figure 3.  $ a = 5 $. (a) Subcritical bifurcation diagram when $ k = 0.8 $. (b) Supercritical bifurcation diagram when $ k = 0.1 $
Figure 4.  $ k = 0.1 $ $ a = 5 $. For $ \tau = 1.05 $, the solution of (19) with initial conditions $ u_0(x, t) = 0.3-0.16\cos x $, $ v_0(x, t) = 0.5-0.16\cos x $ converges to a periodic solution
Figure 5.  $ k = 0.1 $, $ a = 5 $. For $ \tau = 1.05 $, the solution of (19) with initial conditions $ u_0(x, t) = 10.3-0.16\cos x $, $ v_0(x, t) = 10.5-0.16\cos x $ diverges to infinity
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