July  2009, 12(1): 77-108. doi: 10.3934/dcdsb.2009.12.77

Particle, kinetic and fluid models for phototaxis

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

2. 

Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742

Received  March 2008 Revised  February 2009 Published  May 2009

In this work we derive a hierarchy of new mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. These models are based on recent experiments suggesting that the motion of such bacteria depends on the individual bacteria, on group dynamics, and on the interaction between bacteria and their environment. Our first model is a collisionless interacting particle system in which we follow the location of the bacteria, their velocity, and their internal excitation (a parameter whose role is assumed to be related to communication between bacteria). In this model, the light source acts as an external force. The resulting particle system is an extension of the Cucker-Smale flocking model. We prove that when all particles are fully excited, their asymptotic velocity tends to an identical (pre-determined) terminal velocity. Our second model is a kinetic model for the one-particle distribution function that includes an internal variable representing the excitation level. The kinetic model is a Vlasov-type equation that is derived from the particle system using the BBGKY hierarchy and molecular chaos assumption. Since bacteria tend to move in areas that were previously traveled by other bacteria, a surface memory effect is added to the kinetic model as a turning operator that accounts for the collisions between bacteria and the environment. The third and final model is derived as a formal macroscopic limit of the kinetic model. It is shown to be the Vlasov-McKean equation coupled with a reaction-diffusion equation.
Citation: Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77
[1]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[2]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[3]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[4]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[5]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[6]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[7]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253

[8]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027

[9]

Jan Haskovec. Cucker-Smale model with finite speed of information propagation: well-posedness, flocking and mean-field limit. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022033

[10]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008

[11]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[12]

Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164

[13]

Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic and Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031

[14]

Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim, Woojoo Shim. Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057

[15]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195

[16]

Huyên Pham. Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 7-. doi: 10.1186/s41546-016-0008-x

[17]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[18]

Chunyin Jin. Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022023

[19]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[20]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (132)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]