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

Tarik Mohammed Touaoula; Mohammed Nor Frioui; Nikolay Bessonov; Vitaly Volpert

doi:10.3934/dcdss.2020193

Article Contents

2020, Volume 13, Issue 9: 2425-2442. Doi: 10.3934/dcdss.2020193

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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
^* Corresponding author: Vitaly Volpert

Received: October 2018

Revised: June 2019

Early access: December 2019

Published: June 2020

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Abstract

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.

Keywords:

Mathematics Subject Classification: 34K20, 37L15, 92C37.

Citation:

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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)

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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)

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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)

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Figure 3. Maximum maps for one-maximum (1M) symmetric mode, $ \tau = 1.1 $, D = 0.0004 (left) and D = 0.00038 (right)

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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)

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Figure 5. Maximum maps for 2M symmetric mode, $ \tau = 1.1 $, D = 0.000195 (left) and D = 0.000175 (right)

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Figure 6. Maximum maps for 1M symmetric mode, $ \tau = 1.1, D = 0.0005 $, $ \delta = 10^{-6} $ (left) and $ \delta = 10^{-5} $ (right)

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Figure 7. Maximum maps for 1M spinning mode, $ \tau = 1.1, D = 0.0001 $, $ \delta = 10^{-5} $ (left) and $ \delta = 10^{-4} $ (right)

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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)

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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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