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February  2019, 24(2): 487-510. doi: 10.3934/dcdsb.2018183

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang

Received  November 2017 Published  June 2018

Fund Project: The authors are supported the National Natural Science Foundation of China (No.11371112).

We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

Citation: Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
References:
[1]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of turing-hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

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J. Carr, Applications of Centre Manifold Theory, vol. 35, Springer-Verlag, New York-Berlin, 1981. doi: 10.1007/978-1-4612-5929-9.  Google Scholar

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V. Castets, E. Dulos, J. Boissonade and P. D. Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953.  Google Scholar

[4]

S. S. Chen, Y. Lou and J. J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359, arXiv: 1706.02087. doi: 10.1016/j.jde.2018.01.008.  Google Scholar

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T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.  Google Scholar

[6]

T. Faria and L. T. Magalhaes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations, 122 (1995), 181-200.  doi: 10.1006/jdeq.1995.1144.  Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

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K. P. Hadeler and S. G. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 95-105.  doi: 10.3934/dcdsb.2007.8.95.  Google Scholar

[9]

J. R. HuangZ. H. Liu and S. G. Ruan, Bifurcation and temporal periodic patterns in a plant-pollinator model with diffusion and time delay effects, J. Biol. Dyn., 11 (2017), 138-159.  doi: 10.1080/17513758.2016.1181802.  Google Scholar

[10]

R. E. KooijJ. T. Arus and A. G. Embid, Limit cycles in the holling-tanner model, Publ. Mat., 41 (1997), 149-167.  doi: 10.5565/PUBLMAT_41197_09.  Google Scholar

[11]

I. Lengyel and I. R. Epstein, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.  Google Scholar

[12]

W. T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[13]

X. D. LinJ. W. H. So and J. H. Wu, Centre manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh, 122 (1992), 237-254.  doi: 10.1017/S0308210500021090.  Google Scholar

[14]

P. K. MainiK. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.  doi: 10.1039/a702602a.  Google Scholar

[15]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.  Google Scholar

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J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.  Google Scholar

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Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped turing patterns, Nature, 352 (1991), 610-612.  doi: 10.1038/352610a0.  Google Scholar

[18]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[19]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9.  Google Scholar

[20]

A. Rovinsky and M. Menzinger, Interaction of turing and hopf bifurcations in chemical systems, Phys. Rev. A, 46 (1992), 6315-6322.  doi: 10.1103/PhysRevA.46.6315.  Google Scholar

[21]

E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar

[22]

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

[23]

H. B. Shi and S. G. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.  Google Scholar

[24]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.2307/1997265.  Google Scholar

[25]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[26]

H. B. Wang and W. H. Jiang, Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 368 (2010), 9-18.  doi: 10.1016/j.jmaa.2010.03.012.  Google Scholar

[27]

A. D. WitD. LimaG. Dewel and P. Borckmans, Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.  doi: 10.1103/PhysRevE.54.261.  Google Scholar

[28]

J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[29]

J. H. Wu and X. F. Zou, Traveling wave fronts of Reaction-Diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[30]

X. F. Xu and J. J. Wei, Bifurcation analysis of a spruce budworm model with diffusion and physiological structures, J. Differential Equations, 262 (2017), 5206-5230.  doi: 10.1016/j.jde.2017.01.023.  Google Scholar

[31]

R. Yang and Y. L. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 31 (2016), 356-387.  doi: 10.1016/j.nonrwa.2016.02.006.  Google Scholar

[32]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

References:
[1]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of turing-hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

[2]

J. Carr, Applications of Centre Manifold Theory, vol. 35, Springer-Verlag, New York-Berlin, 1981. doi: 10.1007/978-1-4612-5929-9.  Google Scholar

[3]

V. Castets, E. Dulos, J. Boissonade and P. D. Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953.  Google Scholar

[4]

S. S. Chen, Y. Lou and J. J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359, arXiv: 1706.02087. doi: 10.1016/j.jde.2018.01.008.  Google Scholar

[5]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.  Google Scholar

[6]

T. Faria and L. T. Magalhaes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations, 122 (1995), 181-200.  doi: 10.1006/jdeq.1995.1144.  Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[8]

K. P. Hadeler and S. G. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 95-105.  doi: 10.3934/dcdsb.2007.8.95.  Google Scholar

[9]

J. R. HuangZ. H. Liu and S. G. Ruan, Bifurcation and temporal periodic patterns in a plant-pollinator model with diffusion and time delay effects, J. Biol. Dyn., 11 (2017), 138-159.  doi: 10.1080/17513758.2016.1181802.  Google Scholar

[10]

R. E. KooijJ. T. Arus and A. G. Embid, Limit cycles in the holling-tanner model, Publ. Mat., 41 (1997), 149-167.  doi: 10.5565/PUBLMAT_41197_09.  Google Scholar

[11]

I. Lengyel and I. R. Epstein, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.  Google Scholar

[12]

W. T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[13]

X. D. LinJ. W. H. So and J. H. Wu, Centre manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh, 122 (1992), 237-254.  doi: 10.1017/S0308210500021090.  Google Scholar

[14]

P. K. MainiK. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.  doi: 10.1039/a702602a.  Google Scholar

[15]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.  Google Scholar

[16]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.  Google Scholar

[17]

Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped turing patterns, Nature, 352 (1991), 610-612.  doi: 10.1038/352610a0.  Google Scholar

[18]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[19]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9.  Google Scholar

[20]

A. Rovinsky and M. Menzinger, Interaction of turing and hopf bifurcations in chemical systems, Phys. Rev. A, 46 (1992), 6315-6322.  doi: 10.1103/PhysRevA.46.6315.  Google Scholar

[21]

E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar

[22]

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

[23]

H. B. Shi and S. G. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.  Google Scholar

[24]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.2307/1997265.  Google Scholar

[25]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[26]

H. B. Wang and W. H. Jiang, Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 368 (2010), 9-18.  doi: 10.1016/j.jmaa.2010.03.012.  Google Scholar

[27]

A. D. WitD. LimaG. Dewel and P. Borckmans, Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.  doi: 10.1103/PhysRevE.54.261.  Google Scholar

[28]

J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[29]

J. H. Wu and X. F. Zou, Traveling wave fronts of Reaction-Diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[30]

X. F. Xu and J. J. Wei, Bifurcation analysis of a spruce budworm model with diffusion and physiological structures, J. Differential Equations, 262 (2017), 5206-5230.  doi: 10.1016/j.jde.2017.01.023.  Google Scholar

[31]

R. Yang and Y. L. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 31 (2016), 356-387.  doi: 10.1016/j.nonrwa.2016.02.006.  Google Scholar

[32]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

Figure 1.  The spatially non-homogeneous periodic attractors $H = (H_1, H_2)$, under the Case 4. Here $\rho_4 = 0.2, ~\phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $w_4 = 1$, $l = 1$, $n_2 = 3$
Figure 2.  The spatially non-homogeneous quasi-periodic attractors $H = (H_1, H_2)$, under the case 5. Here $\rho_4 = 0.2, \rho_5 = 0.1, \phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $h_5 = (0.2, 0.5)^{\mathrm{{T}}}$, $w_5 = 1$, $\varpi = 0.5$, $l = 1$, $n_2 = 3$
Figure 3.  (a) Bifurcation sets in $(\alpha_1, \alpha_2)$ plane. (b) Phase portraits in $D_1-D_6$
Figure 4.  Two spatially inhomogeneous periodic solutions coexist in $D_3$, with $(\alpha_1, \alpha_2) = (0.05, -0.33)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$
Figure 5.  Two spatially inhomogeneous steady state solutions coexist in $D_5$, with $(\alpha_1, \alpha_2) = (-0.1, -0.4)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$
Table 1.  The correspondence between the planner and original system
Planar system (29) The original system (1)
$E_1$ Constant steady state $(0, 0)$
$E_2$ Spatially homogeneous periodic solution
$E_3$ Non-constant steady state
$E_4$ Spatially non-homogeneous periodic solution
Periodic solution Spatially non-homogeneous quasi-periodic solution
Planar system (29) The original system (1)
$E_1$ Constant steady state $(0, 0)$
$E_2$ Spatially homogeneous periodic solution
$E_3$ Non-constant steady state
$E_4$ Spatially non-homogeneous periodic solution
Periodic solution Spatially non-homogeneous quasi-periodic solution
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