Stabilization and pattern formation in chemotaxis models with acceleration and logistic source
We consider the following chemotaxis-growth system with an acceleration assumption, $ \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \mathbf{w} \right)+\gamma\left({u-u^\alpha}\right), & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \m...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2023093?viewType=HTML |
| _version_ | 1811317149343416320 |
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| author | Chunlai Mu Weirun Tao |
| author_facet | Chunlai Mu Weirun Tao |
| author_sort | Chunlai Mu |
| collection | DOAJ |
| description | We consider the following chemotaxis-growth system with an acceleration assumption,
$ \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \mathbf{w} \right)+\gamma\left({u-u^\alpha}\right), & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \mathbf{w}_t= \Delta \mathbf{w} - \mathbf{w} +\chi\nabla v, & x\in\Omega,\ t>0, \end{cases} \end{align*} $
under the homogeneous Neumann boundary condition for $ u, v $ and the homogeneous Dirichlet boundary condition for $ \mathbf{w} $ in a smooth bounded domain $ \Omega\subset \mathbb{R}^{n} $ ($ n\geq1 $) with given parameters $ \chi > 0 $, $ \gamma\geq0 $ and $ \alpha > 1 $. It is proved that for reasonable initial data with either $ n\leq3 $, $ \gamma\geq0 $, $ \alpha > 1 $ or $ n\geq4, \ \gamma > 0, \ \alpha > \frac12+\frac n4 $, the system admits global bounded solutions, which significantly differs from the classical chemotaxis model that may have blow-up solutions in two and three dimensions. For given $ \gamma $ and $ \alpha $, the obtained global bounded solutions are shown to convergence exponentially to the spatially homogeneous steady state $ (m, m, \bf 0 $) in the large time limit for appropriately small $ \chi $, where $ m = \frac1{|\Omega|} \int_\Omega u_0(x) $ if $ \gamma = 0 $ and $ m = 1 $ if $ \gamma > 0 $. Outside the stable parameter regime, we conduct linear analysis to specify possible patterning regimes. In weakly nonlinear parameter regimes, with a standard perturbation expansion approach, we show that the above asymmetric model can generate pitchfork bifurcations which occur generically in symmetric systems. Moreover, our numerical simulations demonstrate that the model can generate rich aggregation patterns, including stationary, single merging aggregation, merging and emerging chaotic, and spatially inhomogeneous time-periodic. Some open questions for further research are discussed.
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| first_indexed | 2024-04-13T12:02:57Z |
| format | Article |
| id | doaj.art-ad10523cd2ec4be5a930c95546b218fa |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-13T12:02:57Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-ad10523cd2ec4be5a930c95546b218fa2022-12-22T02:47:44ZengAIMS PressMathematical Biosciences and Engineering1551-00182023-01-012022011203810.3934/mbe.2023093Stabilization and pattern formation in chemotaxis models with acceleration and logistic sourceChunlai Mu0Weirun Tao11. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong KongWe consider the following chemotaxis-growth system with an acceleration assumption, $ \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \mathbf{w} \right)+\gamma\left({u-u^\alpha}\right), & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \mathbf{w}_t= \Delta \mathbf{w} - \mathbf{w} +\chi\nabla v, & x\in\Omega,\ t>0, \end{cases} \end{align*} $ under the homogeneous Neumann boundary condition for $ u, v $ and the homogeneous Dirichlet boundary condition for $ \mathbf{w} $ in a smooth bounded domain $ \Omega\subset \mathbb{R}^{n} $ ($ n\geq1 $) with given parameters $ \chi > 0 $, $ \gamma\geq0 $ and $ \alpha > 1 $. It is proved that for reasonable initial data with either $ n\leq3 $, $ \gamma\geq0 $, $ \alpha > 1 $ or $ n\geq4, \ \gamma > 0, \ \alpha > \frac12+\frac n4 $, the system admits global bounded solutions, which significantly differs from the classical chemotaxis model that may have blow-up solutions in two and three dimensions. For given $ \gamma $ and $ \alpha $, the obtained global bounded solutions are shown to convergence exponentially to the spatially homogeneous steady state $ (m, m, \bf 0 $) in the large time limit for appropriately small $ \chi $, where $ m = \frac1{|\Omega|} \int_\Omega u_0(x) $ if $ \gamma = 0 $ and $ m = 1 $ if $ \gamma > 0 $. Outside the stable parameter regime, we conduct linear analysis to specify possible patterning regimes. In weakly nonlinear parameter regimes, with a standard perturbation expansion approach, we show that the above asymmetric model can generate pitchfork bifurcations which occur generically in symmetric systems. Moreover, our numerical simulations demonstrate that the model can generate rich aggregation patterns, including stationary, single merging aggregation, merging and emerging chaotic, and spatially inhomogeneous time-periodic. Some open questions for further research are discussed. https://www.aimspress.com/article/doi/10.3934/mbe.2023093?viewType=HTMLchemotaxisaccelerationstabilizationamplitude equationpattern formation |
| spellingShingle | Chunlai Mu Weirun Tao Stabilization and pattern formation in chemotaxis models with acceleration and logistic source Mathematical Biosciences and Engineering chemotaxis acceleration stabilization amplitude equation pattern formation |
| title | Stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| title_full | Stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| title_fullStr | Stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| title_full_unstemmed | Stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| title_short | Stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| title_sort | stabilization and pattern formation in chemotaxis models with acceleration and logistic source |
| topic | chemotaxis acceleration stabilization amplitude equation pattern formation |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2023093?viewType=HTML |
| work_keys_str_mv | AT chunlaimu stabilizationandpatternformationinchemotaxismodelswithaccelerationandlogisticsource AT weiruntao stabilizationandpatternformationinchemotaxismodelswithaccelerationandlogisticsource |