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## 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, doi: 10.3934/dcdsb.2019118
##### 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. [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. [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. [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. [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. [6] J. D. Ferreira, A. 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. [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. [8] S. Guo, Y. 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. [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. [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. [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. [12] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981. [13] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [14] A. V. Ion, On the Bautin bifurcation for systems of delay differential equations, Acta Univ. Apulensis, 8 (2004), 235-246. [15] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004. doi: 10.1007%2F978-1-4757-3978-7. [16] X. Lin, J. 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. [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. [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. [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. [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. [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. [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. [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. [24] Y. Song, T. 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. [25] Y. Su, J. 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. [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. [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. [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. [29] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. [30] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007%2F978-1-4612-4050-1. [31] X. Yang, M. Yang, H. 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. [32] F. Yi, J. 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. [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.

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. [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. [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. [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. [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. [6] J. D. Ferreira, A. 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. [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. [8] S. Guo, Y. 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. [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. [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. [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. [12] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981. [13] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [14] A. V. Ion, On the Bautin bifurcation for systems of delay differential equations, Acta Univ. Apulensis, 8 (2004), 235-246. [15] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004. doi: 10.1007%2F978-1-4757-3978-7. [16] X. Lin, J. 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. [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. [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. [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. [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. [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. [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. [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. [24] Y. Song, T. 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. [25] Y. Su, J. 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. [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. [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. [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. [29] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. [30] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007%2F978-1-4612-4050-1. [31] X. Yang, M. Yang, H. 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. [32] F. Yi, J. 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. [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.
The Bautin bifurcation diagram near the equilibrium for a) $l_2(0)<0$ and b) $l_2(0)>0$
$a = 5$. a) the first Lyapunov coefficient $l_1$ varies with respect to $k$. b) the Bautin bifurcation diagram near the Bautin point $(k^\ast, \tau^\ast) = (0.3075, 0.6543)$
$a = 5$. (a) Subcritical bifurcation diagram when $k = 0.8$. (b) Supercritical bifurcation diagram when $k = 0.1$
$k = 0.1$ $a = 5$. 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
$k = 0.1$, $a = 5$. 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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