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

  • * Corresponding author: Vitaly Volpert

    * Corresponding author: Vitaly Volpert
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  • 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.

    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)

    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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