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doi: 10.3934/dcdss.2021150
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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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021150
References:
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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.  Google Scholar

[2]

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

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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.   Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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[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.  Google Scholar

[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.   Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[4]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[15]

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

[16]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.   Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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