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Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model

  • * Corresponding author: Moustafa Ibrahim

    * Corresponding author: Moustafa Ibrahim 
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  • Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.

    Mathematics Subject Classification: Primary: 35B40, 35K57, 35K55, 35K65, 65M08; Secondary: 70K50, 92C15.

    Citation:

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  • Figure 1.  Unstructured triangular mesh for the space domain $ {\Omega} = {\left({0,1}\right)}\times{\left({0,1}\right)} $ with 14336 acute angle triangles

    Figure 2.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the chemosensitivity strength $ \zeta $ increases beyond the critical value $ \zeta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi

    Figure 3.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $

    Figure 4.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero. 2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)

    Figure 5.  First row from left to right: Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 2.5 $, $ t = 325 $, and $ t = 997.5 $. Second row from left to right: Evolution of the heterogeneous stationary solutions at the same moments as for the evolution of $ u{\left({ {\mathbf{x}},t}\right)} $

    Figure 6.  Similarities of patterns between the nonlinear evolution $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)

    Figure 7.  Time evolution of the difference in $ {L^{2}} $ between $ u{\left({ {\mathbf{x}},t}\right)} $ and the heterogeneous solution

    Figure 8.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the death rate $ \beta $ decreases below the critical value $ \beta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi, and pattern formation can be expected

    Figure 9.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $

    Figure 10.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero.2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)

    Figure 11.  First row from left to right. Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 10 $, $ t = 70 $, and $ t = 750 $. Second row from left to right. Evolution of the heterogeneous stationary solutions at the same moments as for $ u{\left({ {\mathbf{x}},t}\right)} $

    Figure 12.  Similarities of patterns between the nonlinear evolution $ u({\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)

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