March  2022, 15(3): 535-554. doi: 10.3934/dcdss.2021150

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

1. 

Çankaya University, Department of Mathematics, Eskişehir Yolu 29.km, Yukarıyurtçu Mahallesi, Mimar Sinan Caddesi, No: 4, 06790, Ankara, Turkey

2. 

TOBB University of Economics and Technology, Department of Mathematics, Söğütözü Caddesi, No: 43, 06560, Ankara, Turkey

3. 

Polytechnic University of Marche, Department of Management, Piazza Martelli 8, 60121, Ancona, Italy

* Corresponding author: Huseyin Merdan

Received  January 2020 Revised  February 2021 Published  March 2022 Early access  November 2021

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.

Citation: Şeyma Bılazeroğlu, Huseyin Merdan, Luca Guerrini. Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 535-554. doi: 10.3934/dcdss.2021150
References:
[1]

H. AkkocaoğluH. Merdan and C. Çelik, Hopf bifurcation analysis of a general non-linear differential equation with delay, J. Comput. Appl. Math., 237 (2013), 565-575.  doi: 10.1016/j.cam.2012.06.029.

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L. J. S. Allen, An Introduction to Mathematical Biology, Pearson-Prentice Hall, New Jersey, 2007.

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A. A. Andronov and A. Witt, Sur la theórie mathematiques des autooscillations, C. R. Acad. Sci. Paris, 237 (1930), 256-258. 

[4]

B. Balachandran, T. Kalmar-Nagy and D. E. Gilsinn, Delay Differential Equations: Recent Advances and New Directions, Springer, New York, 2009.

[5] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. 
[6]

P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Sys., 12 (2013), 1847-1888.  doi: 10.1137/120887898.

[7]

Ş. Bilazeroğlu and H. Merdan, Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications, Chaos, Solitons Fractals, 142 (2021), 110391.  doi: 10.1016/j.chaos.2020.110391.

[8]

C. Çelik and H. Merdan, Hopf bifurcation analysis of a system of coupled delayed-differential equations, Appl. Math. Comput., 219 (2013), 6605-6617.  doi: 10.1016/j.amc.2012.12.063.

[9]

N. Chafee, A bifurcation problem for functional differential equation of finitely retarded type, J. Math. Anal. Appl., 35 (1971), 312-348.  doi: 10.1016/0022-247X(71)90221-6.

[10]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.

[11]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. 

[12]

P. De KepperV. CastetsE. Dulos and J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D, 49 (1991), 161-169.  doi: 10.1016/0167-2789(91)90204-M.

[13]

L. Du and M. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.  doi: 10.1016/j.jmaa.2010.02.002.

[14] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics, Oxford University Press, Oxford, 1998.  doi: 10.1093/oso/9780195096705.001.0001.
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J. K. Hale, Theory of Functional Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York-Heidelberg, 1977.

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J. K. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981. 
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E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Nat. Kl., 95 (1943), 3-22. 

[19]

J. JangW. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[20]

J. JinJ. ShiJ. Wei and F. Yi, Bifurcations of patterned soltions in the diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.

[21]

E. Karaoglu and H. Merdan, Hopf bifurcation analysis for a ratio-dependent predator-prey system involving two delays, ANZIAM J., 55 (2014), 214-231.  doi: 10.1017/S1446181114000054.

[22]

E. Karaoglu and H. Merdan, Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays, Chaos, Soliton Fractals, 68 (2014), 159-168.  doi: 10.1016/j.chaos.2014.07.011.

[23]

Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, Inc., Boston, MA, 1993.

[24]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[25]

I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[26]

I. Lengyel and I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion system, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979.  doi: 10.1073/pnas.89.9.3977.

[27]

B. Li and M. Wang, Diffusion-driven instability and Hopf bifurcation in Brusselator system, Appl. Math. Mech. (English Ed.), 29 (2008), 825-832.  doi: 10.1007/s10483-008-0614-y.

[28]

Z. P. Ma, Stability and Hopf bifurcation for a three-component reaction-diffusion population model with delay effect, Appl. Math. Model., 37 (2013), 5984-6007.  doi: 10.1016/j.apm.2012.12.012.

[29]

X-C. Mao and H-Y. Hu, Hopf bifurcation analysis of a four-neuron network with multiple time delays, Nonliear Dynam., 55 (2009), 95-112.  doi: 10.1007/s11071-008-9348-0.

[30]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976.

[31]

H. Merdan and Ş. Kayan, Hopf bifurcations in Lengyel-Epstein rection-diffusion model with discrete time delay, Nonliear Dynam., 79 (2015), 1757-1770.  doi: 10.1007/s11071-014-1772-8.

[32]

H. Merdan and Ş. Kayan, Delay effects on the dynamics of the lengyel-epstein reaction-diffusion model, Mathematical Modelling and Applications in Nonlinear Dynamics, Nonlinear Syst. Complex., 14 (2016), 125-160. 

[33]

J. D. Murray, Mathematical Biology, Springer, New York, 2002.

[34]

W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[35]

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.

[36]

S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natur. Resource Modeling, 11 (1998), 131-142.  doi: 10.1111/j.1939-7445.1998.tb00304.x.

[37]

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

[38]

J. Wu, Theory and Applications of Partial Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

C. Xu and Y. Shao, Bifurcations in a predator-prey model with discrete and disributed time delay, Nonliear Dynam., 67 (2012), 2207-2223.  doi: 10.1007/s11071-011-0140-1.

[40]

R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947.

[41]

F. YiJ. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.

[42]

F. YiJ. Wei and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reactio-diffusion system, Appl. Math. Lett., 22 (2009), 52-55.  doi: 10.1016/j.aml.2008.02.003.

[43]

G. ZangY. Shen and B. Chen, Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonliear Dynam., 73 (2013), 2119-2131.  doi: 10.1007/s11071-013-0928-2.

show all references

References:
[1]

H. AkkocaoğluH. Merdan and C. Çelik, Hopf bifurcation analysis of a general non-linear differential equation with delay, J. Comput. Appl. Math., 237 (2013), 565-575.  doi: 10.1016/j.cam.2012.06.029.

[2]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson-Prentice Hall, New Jersey, 2007.

[3]

A. A. Andronov and A. Witt, Sur la theórie mathematiques des autooscillations, C. R. Acad. Sci. Paris, 237 (1930), 256-258. 

[4]

B. Balachandran, T. Kalmar-Nagy and D. E. Gilsinn, Delay Differential Equations: Recent Advances and New Directions, Springer, New York, 2009.

[5] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. 
[6]

P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Sys., 12 (2013), 1847-1888.  doi: 10.1137/120887898.

[7]

Ş. Bilazeroğlu and H. Merdan, Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications, Chaos, Solitons Fractals, 142 (2021), 110391.  doi: 10.1016/j.chaos.2020.110391.

[8]

C. Çelik and H. Merdan, Hopf bifurcation analysis of a system of coupled delayed-differential equations, Appl. Math. Comput., 219 (2013), 6605-6617.  doi: 10.1016/j.amc.2012.12.063.

[9]

N. Chafee, A bifurcation problem for functional differential equation of finitely retarded type, J. Math. Anal. Appl., 35 (1971), 312-348.  doi: 10.1016/0022-247X(71)90221-6.

[10]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8.

[11]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. 

[12]

P. De KepperV. CastetsE. Dulos and J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D, 49 (1991), 161-169.  doi: 10.1016/0167-2789(91)90204-M.

[13]

L. Du and M. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.  doi: 10.1016/j.jmaa.2010.02.002.

[14] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics, Oxford University Press, Oxford, 1998.  doi: 10.1093/oso/9780195096705.001.0001.
[15]

J. K. Hale, Theory of Functional Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York-Heidelberg, 1977.

[16]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981. 
[18]

E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Nat. Kl., 95 (1943), 3-22. 

[19]

J. JangW. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[20]

J. JinJ. ShiJ. Wei and F. Yi, Bifurcations of patterned soltions in the diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.

[21]

E. Karaoglu and H. Merdan, Hopf bifurcation analysis for a ratio-dependent predator-prey system involving two delays, ANZIAM J., 55 (2014), 214-231.  doi: 10.1017/S1446181114000054.

[22]

E. Karaoglu and H. Merdan, Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays, Chaos, Soliton Fractals, 68 (2014), 159-168.  doi: 10.1016/j.chaos.2014.07.011.

[23]

Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, Inc., Boston, MA, 1993.

[24]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[25]

I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[26]

I. Lengyel and I. R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion system, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979.  doi: 10.1073/pnas.89.9.3977.

[27]

B. Li and M. Wang, Diffusion-driven instability and Hopf bifurcation in Brusselator system, Appl. Math. Mech. (English Ed.), 29 (2008), 825-832.  doi: 10.1007/s10483-008-0614-y.

[28]

Z. P. Ma, Stability and Hopf bifurcation for a three-component reaction-diffusion population model with delay effect, Appl. Math. Model., 37 (2013), 5984-6007.  doi: 10.1016/j.apm.2012.12.012.

[29]

X-C. Mao and H-Y. Hu, Hopf bifurcation analysis of a four-neuron network with multiple time delays, Nonliear Dynam., 55 (2009), 95-112.  doi: 10.1007/s11071-008-9348-0.

[30]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976.

[31]

H. Merdan and Ş. Kayan, Hopf bifurcations in Lengyel-Epstein rection-diffusion model with discrete time delay, Nonliear Dynam., 79 (2015), 1757-1770.  doi: 10.1007/s11071-014-1772-8.

[32]

H. Merdan and Ş. Kayan, Delay effects on the dynamics of the lengyel-epstein reaction-diffusion model, Mathematical Modelling and Applications in Nonlinear Dynamics, Nonlinear Syst. Complex., 14 (2016), 125-160. 

[33]

J. D. Murray, Mathematical Biology, Springer, New York, 2002.

[34]

W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[35]

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.

[36]

S. Ruan, Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis, Natur. Resource Modeling, 11 (1998), 131-142.  doi: 10.1111/j.1939-7445.1998.tb00304.x.

[37]

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

[38]

J. Wu, Theory and Applications of Partial Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

C. Xu and Y. Shao, Bifurcations in a predator-prey model with discrete and disributed time delay, Nonliear Dynam., 67 (2012), 2207-2223.  doi: 10.1007/s11071-011-0140-1.

[40]

R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947.

[41]

F. YiJ. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.

[42]

F. YiJ. Wei and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reactio-diffusion system, Appl. Math. Lett., 22 (2009), 52-55.  doi: 10.1016/j.aml.2008.02.003.

[43]

G. ZangY. Shen and B. Chen, Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonliear Dynam., 73 (2013), 2119-2131.  doi: 10.1007/s11071-013-0928-2.

Figure 1.  When $ \tau_{2} = 0.005 < \tau_{2_0} $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 2.  When $ \tau_{2} = \tau_{2_0} \doteq 0.0135 $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 3.  When $ \tau_{2} = 0.0140\in \lbrack 0.0135,0.0135+\varepsilon) $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 4.  When $ \tau_{2} = 0.0145\in \lbrack 0.0135,0.0135+\varepsilon) $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 5.  When $ \tau_{2} = 0.02 > \tau_{2_0} $ and $ \tau_{1} = 0.0350 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 6.  When $ \tau_{1} = 0.01 < \tau_{1_0} $ and $ \tau_{2} = 0.02 $, they are phase portrait of (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 7.  When $ \tau_{1} = \tau_{1_0} \doteq 0.0287 $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 8.  When $ \tau_{1} = 0.0288\in \lbrack 0.0287,0.0287+\varepsilon) $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 9.  When $ \tau_{1} = 0.0290\in \lbrack 0.0287,0.0287+\varepsilon) $ and $ \tau_{2} = 0.02 $, they are portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
Figure 10.  When $ \tau_{1} = 0.04 > \tau_{1_0} $ and $ \tau_{2} = 0.02 $, they are phase portrait of system (54) (left), trajectory of activator density versus time (middle), trajectory of inhibitor density versus time (right), respectively. The initial conditions are $ (u_{0},v_{0}) = (3.25,9.75) $ for each simulation
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Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751

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Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

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