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doi: 10.3934/dcdss.2020193

Dynamics of solutions of a reaction-diffusion equation with delayed inhibition

1. 

Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Département de Mathématiques, Université Aboubekr Belkaïd Tlemcen, 13000 Tlemcen, Algeria

2. 

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia

3. 

Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

4. 

Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France

5. 

INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France

6. 

Marchuk Institute of Numerical Mathematics of the RAS, ul. Gubkina 8, 119333 Moscow, Russian Federation

* Corresponding author: Vitaly Volpert

Received  October 2018 Revised  June 2019 Published  December 2019

Reaction-diffusion equation with a logistic production term and a delayed inhibition term is studied. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.

Citation: Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020193
References:
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L. BerezanskyE. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.  doi: 10.1016/j.amc.2012.12.043.  Google Scholar

[2]

N. BessonovG. BocharovT. M. TouaoulaS. Trofimchuk and V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.  doi: 10.3934/dcdsb.2019085.  Google Scholar

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N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

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N. BessonovN. ReinbergM. Banerjee and V. Volpert, The origin of species by means of mathematical modelling, Acta Bioteoretica, 66 (2018), 333-344.  doi: 10.1007/s10441-018-9328-9.  Google Scholar

[5]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PLoS ONE, (2016). doi: 10.1371/journal.pone.0168576.  Google Scholar

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G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Internat. J. Parallel Emergent Distributed Syst., (2017), 341–355. doi: 10.1080/17445760.2017.1363203.  Google Scholar

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E. Braverman and S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.  doi: 10.3934/dcds.2012.32.2041.  Google Scholar

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K. Deng and Y. Wu, On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.  doi: 10.1016/j.aml.2015.06.013.  Google Scholar

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W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar

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Z. LingL. Zhang and Z. Lin, Turing pattern formation in a predator-prey system with cross diffusion, Appl. Math. Model., 38 (2014), 5022-5032.  doi: 10.1016/j.apm.2014.04.015.  Google Scholar

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E. LizM. PintoV. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  doi: 10.1090/S0033-569X-05-00951-3.  Google Scholar

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E. LizV. Tkachenko and S. Tromichuk, A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.  doi: 10.1137/S0036141001399222.  Google Scholar

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[15]

J. Mallet-Paret and G. R. Sell., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

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G. Rost and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.  Google Scholar

[20]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[21]

G. Q. SunC. H. WangL. L. ChangY. P. Wu and L. L. Zhen Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.  Google Scholar

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. doi: 10.2307/j.ctv301f9v.  Google Scholar

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[24]

T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191.  Google Scholar

[25]

S. Trofimchuk and V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1190.  doi: 10.1137/17M1115587.  Google Scholar

[26]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems: Translation of Mathematical Monographs, Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994.  Google Scholar

[27]

J. Wu, Theory and applications of partial functional differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[28]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.   Google Scholar

[29]

T. YiY. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.  doi: 10.1007/s00033-012-0224-x.  Google Scholar

[30]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.  doi: 10.1016/j.jde.2008.03.007.  Google Scholar

[31]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

[32]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027.  Google Scholar

[33]

T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dynam. Differential Equations, 25 (2013), 959-979.  doi: 10.1007/s10884-013-9324-3.  Google Scholar

[34]

T. Yi and X. Zou, Dirichlet problem for a delayed reaction-diffusion equation on a semi-infinite interval, J. Dynam. Differential Equations, 28 (2016), 1007-1030.  doi: 10.1007/s10884-015-9457-7.  Google Scholar

[35]

Y. Yuan and X. Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Differential Equations, 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026.  Google Scholar

[36]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.   Google Scholar

show all references

References:
[1]

L. BerezanskyE. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.  doi: 10.1016/j.amc.2012.12.043.  Google Scholar

[2]

N. BessonovG. BocharovT. M. TouaoulaS. Trofimchuk and V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.  doi: 10.3934/dcdsb.2019085.  Google Scholar

[3]

N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

[4]

N. BessonovN. ReinbergM. Banerjee and V. Volpert, The origin of species by means of mathematical modelling, Acta Bioteoretica, 66 (2018), 333-344.  doi: 10.1007/s10441-018-9328-9.  Google Scholar

[5]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PLoS ONE, (2016). doi: 10.1371/journal.pone.0168576.  Google Scholar

[6]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Internat. J. Parallel Emergent Distributed Syst., (2017), 341–355. doi: 10.1080/17445760.2017.1363203.  Google Scholar

[7]

E. Braverman and S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.  doi: 10.3934/dcds.2012.32.2041.  Google Scholar

[8]

K. Deng and Y. Wu, On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.  doi: 10.1016/j.aml.2015.06.013.  Google Scholar

[9]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar

[10]

Z. LingL. Zhang and Z. Lin, Turing pattern formation in a predator-prey system with cross diffusion, Appl. Math. Model., 38 (2014), 5022-5032.  doi: 10.1016/j.apm.2014.04.015.  Google Scholar

[11]

E. LizM. PintoV. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  doi: 10.1090/S0033-569X-05-00951-3.  Google Scholar

[12]

E. Liz and G. Rost, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.  doi: 10.3934/dcds.2009.24.1215.  Google Scholar

[13]

E. LizV. Tkachenko and S. Tromichuk, A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.  doi: 10.1137/S0036141001399222.  Google Scholar

[14]

J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM. J. Math. Anal., 20 (1989), 249-292.  doi: 10.1137/0520019.  Google Scholar

[15]

J. Mallet-Paret and G. R. Sell., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[16]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[17]

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

[18] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar
[19]

G. Rost and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.  Google Scholar

[20]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[21]

G. Q. SunC. H. WangL. L. ChangY. P. Wu and L. L. Zhen Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.  Google Scholar

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. doi: 10.2307/j.ctv301f9v.  Google Scholar

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[24]

T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191.  Google Scholar

[25]

S. Trofimchuk and V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1190.  doi: 10.1137/17M1115587.  Google Scholar

[26]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems: Translation of Mathematical Monographs, Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994.  Google Scholar

[27]

J. Wu, Theory and applications of partial functional differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[28]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.   Google Scholar

[29]

T. YiY. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.  doi: 10.1007/s00033-012-0224-x.  Google Scholar

[30]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.  doi: 10.1016/j.jde.2008.03.007.  Google Scholar

[31]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

[32]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027.  Google Scholar

[33]

T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dynam. Differential Equations, 25 (2013), 959-979.  doi: 10.1007/s10884-013-9324-3.  Google Scholar

[34]

T. Yi and X. Zou, Dirichlet problem for a delayed reaction-diffusion equation on a semi-infinite interval, J. Dynam. Differential Equations, 28 (2016), 1007-1030.  doi: 10.1007/s10884-015-9457-7.  Google Scholar

[35]

Y. Yuan and X. Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Differential Equations, 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026.  Google Scholar

[36]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.   Google Scholar

Figure 1.  Example of numerical simulations of equation (22) with the maximum map of solution (left) and solution $ u(x,t) $ plotted as a function of two variables 3D (right)
Figure 2.  Maximum maps for one-maximum (1M) symmetric mode, $ \tau = 1.1 $, D = 0.001 (left), D = 0.00058 (middle) and D = 0.00045 (right)
Figure 8.  Maximum maps for the functions $ v(x,t) = \cos (t) \cdot \cos (\alpha x - \beta ) $ (left), $ \cos(\alpha x-ct) $ (middle), $ \cos(\alpha |x-x_0|-ct) $ (right)
Figure 3.  Maximum maps for one-maximum (1M) symmetric mode, $ \tau = 1.1 $, D = 0.0004 (left) and D = 0.00038 (right)
Figure 4.  Maximum maps for one-maximum (1M) spinning mode, $ \tau = 1.1 $, D = 0.00031 (left), D = 0.00026 (middle) and D = 0.000125 (right)
Figure 5.  Maximum maps for 2M symmetric mode, $ \tau = 1.1 $, D = 0.000195 (left) and D = 0.000175 (right)
Figure 6.  Maximum maps for 1M symmetric mode, $ \tau = 1.1, D = 0.0005 $, $ \delta = 10^{-6} $ (left) and $ \delta = 10^{-5} $ (right)
Figure 7.  Maximum maps for 1M spinning mode, $ \tau = 1.1, D = 0.0001 $, $ \delta = 10^{-5} $ (left) and $ \delta = 10^{-4} $ (right)
Figure 9.  Maximum maps for the functions $ v(x,t) = \cos (\alpha x - \beta - \epsilon (t) ) $ (left), $ \cos(\alpha |x-x_0| + \epsilon (t) (ct-\beta)) $ (middle), $ \cos (t) \cdot \cos (\alpha |x-x_0|-ct-\epsilon (t)) $ (right)
Figure 10.  Maximum maps for the functions $ v(x,t) = \cos (t) \cdot \cos (\alpha |x-x_0|-cos(ct)+\beta) $ (left), $ \cos(t) \cdot \cos(\alpha x-c_1t) + \cos(\alpha x+c_2 t) $ (middle), $ \cos(t) \cdot \cos(\alpha_1 x+c_1t) + \cos(\alpha x-c_2 t) $ (right)
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