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Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity
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 |
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.
References:
[1] |
M. Baurmann, T. 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. |
[2] |
J. Carr, Applications of Centre Manifold Theory, vol. 35, Springer-Verlag, New York-Berlin, 1981.
doi: 10.1007/978-1-4612-5929-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. R. Huang, Z. 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. |
[10] |
R. E. Kooij, J. 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. |
[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. |
[12] |
W. T. Li, G. 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. |
[13] |
X. D. Lin, J. 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. |
[14] |
P. K. Maini, K. 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. |
[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. |
[16] |
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
A. D. Wit, D. Lima, G. Dewel and P. Borckmans,
Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.
doi: 10.1103/PhysRevE.54.261. |
[28] |
J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[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. |
[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. |
[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. |
[32] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |
show all references
References:
[1] |
M. Baurmann, T. 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. |
[2] |
J. Carr, Applications of Centre Manifold Theory, vol. 35, Springer-Verlag, New York-Berlin, 1981.
doi: 10.1007/978-1-4612-5929-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. R. Huang, Z. 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. |
[10] |
R. E. Kooij, J. 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. |
[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. |
[12] |
W. T. Li, G. 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. |
[13] |
X. D. Lin, J. 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. |
[14] |
P. K. Maini, K. 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. |
[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. |
[16] |
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
A. D. Wit, D. Lima, G. Dewel and P. Borckmans,
Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.
doi: 10.1103/PhysRevE.54.261. |
[28] |
J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[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. |
[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. |
[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. |
[32] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |




Planar system (29) | The original system (1) |
Constant steady state |
|
Spatially homogeneous periodic solution | |
Non-constant steady state | |
Spatially non-homogeneous periodic solution | |
Periodic solution | Spatially non-homogeneous quasi-periodic solution |
Planar system (29) | The original system (1) |
Constant steady state |
|
Spatially homogeneous periodic solution | |
Non-constant steady state | |
Spatially non-homogeneous periodic solution | |
Periodic solution | Spatially non-homogeneous quasi-periodic solution |
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