
Previous Article
Averaging principle for the Schrödinger equations^{†}
 DCDSB Home
 This Issue

Next Article
An analysis of functional curability on HIV infection models with MichaelisMententype immune response and its generalization
Synchronising and nonsynchronising dynamics for a twospecies aggregation model
1.  Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France 
2.  CNRS, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France 
3.  INRIAParisRocquencourt, EPC MAMBA, Domaine de Voluceau, BP105,78153 Le Chesnay Cedex, France 
4.  Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China 
5.  LAGA, UMR 7539, Institut Galilée, Université Paris 13, 99, avenue JeanBaptiste Clément, 93430 Villetaneuse, France 
This paper deals with analysis and numerical simulations of a onedimensional twospecies hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a twospecies population in interaction appearing for instance in bacterial chemotaxis. Blowup of classical solutions occurs in finite time. This raises the question to define measurevalued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measurevalued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measurevalued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blowup of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.
References:
[1] 
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, second ed. , Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar 
[2] 
A. L. Bertozzi and J. Brandman, Finitetime blowup of L^{∞}weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 4565. Google Scholar 
[3] 
F. Bouchut and F. James, Onedimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891933. Google Scholar 
[4] 
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 21732189. Google Scholar 
[5] 
M. CamposPinto, J. A. Carrillo, F. Charles and Y. P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. Google Scholar 
[6] 
J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229271. Google Scholar 
[7] 
J. A. Carrillo, A. Chertock and Y. Huang, A finitevolume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233258. Google Scholar 
[8] 
J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304338. Google Scholar 
[9] 
K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 16811717. Google Scholar 
[10] 
G. Crippa and M. LécureuxMercier, Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523537. Google Scholar 
[11] 
M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 27772808. Google Scholar 
[12] 
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595615. Google Scholar 
[13] 
C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a twospecies chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar 
[14] 
C. Emako, L. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359380. Google Scholar 
[15] 
L. Gosse and F. James, Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 9871015. Google Scholar 
[16] 
D. Helbing, W. Yu and H. Rauhut, Selforganization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177208. Google Scholar 
[17] 
D. D Holm and V. Putkaradze, Aggregation of finitesize particles with variable mobility, Physical Review Letters, 95 (2005), 226106. Google Scholar 
[18] 
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101127. Google Scholar 
[19] 
F. James and N. Vauchelet, Numerical methods for onedimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895916. Google Scholar 
[20] 
F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for onedimensional aggregation equations, Discrete Contin. Dyn. Syst., 36 (2016), 13551382. Google Scholar 
[21] 
T. Liu, M. L. K. Langston, D. Li, J. M. Pigga, C. Pichon, A. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 15901592. Google Scholar 
[22] 
A. Mackey, T. Kolokolnikov and A. Bertozzi, Twospecies particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 14111436. Google Scholar 
[23] 
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 1325913263. Google Scholar 
[24] 
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263298. Google Scholar 
[25] 
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533561. Google Scholar 
[26] 
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), SpringerVerlag, New York, 1998, Applications. Google Scholar 
[27] 
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar 
[28] 
K. SznajdWeron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 11571165. Google Scholar 
[29] 
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. Google Scholar 
[30] 
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, SpringerVerlag, Berlin, 2009. Google Scholar 
[31] 
J. H. Von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31pp. Google Scholar 
show all references
References:
[1] 
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, second ed. , Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar 
[2] 
A. L. Bertozzi and J. Brandman, Finitetime blowup of L^{∞}weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 4565. Google Scholar 
[3] 
F. Bouchut and F. James, Onedimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891933. Google Scholar 
[4] 
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 21732189. Google Scholar 
[5] 
M. CamposPinto, J. A. Carrillo, F. Charles and Y. P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. Google Scholar 
[6] 
J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229271. Google Scholar 
[7] 
J. A. Carrillo, A. Chertock and Y. Huang, A finitevolume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233258. Google Scholar 
[8] 
J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304338. Google Scholar 
[9] 
K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 16811717. Google Scholar 
[10] 
G. Crippa and M. LécureuxMercier, Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523537. Google Scholar 
[11] 
M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 27772808. Google Scholar 
[12] 
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595615. Google Scholar 
[13] 
C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a twospecies chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar 
[14] 
C. Emako, L. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359380. Google Scholar 
[15] 
L. Gosse and F. James, Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 9871015. Google Scholar 
[16] 
D. Helbing, W. Yu and H. Rauhut, Selforganization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177208. Google Scholar 
[17] 
D. D Holm and V. Putkaradze, Aggregation of finitesize particles with variable mobility, Physical Review Letters, 95 (2005), 226106. Google Scholar 
[18] 
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101127. Google Scholar 
[19] 
F. James and N. Vauchelet, Numerical methods for onedimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895916. Google Scholar 
[20] 
F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for onedimensional aggregation equations, Discrete Contin. Dyn. Syst., 36 (2016), 13551382. Google Scholar 
[21] 
T. Liu, M. L. K. Langston, D. Li, J. M. Pigga, C. Pichon, A. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 15901592. Google Scholar 
[22] 
A. Mackey, T. Kolokolnikov and A. Bertozzi, Twospecies particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 14111436. Google Scholar 
[23] 
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 1325913263. Google Scholar 
[24] 
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263298. Google Scholar 
[25] 
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533561. Google Scholar 
[26] 
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), SpringerVerlag, New York, 1998, Applications. Google Scholar 
[27] 
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar 
[28] 
K. SznajdWeron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 11571165. Google Scholar 
[29] 
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. Google Scholar 
[30] 
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, SpringerVerlag, Berlin, 2009. Google Scholar 
[31] 
J. H. Von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31pp. Google Scholar 
[1] 
Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a twospecies kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359380. doi: 10.3934/krm.2015.8.359 
[2] 
Alexander Kurganov, Mária LukáčováMedvidová. Numerical study of twospecies chemotaxis models. Discrete & Continuous Dynamical Systems  B, 2014, 19 (1) : 131152. doi: 10.3934/dcdsb.2014.19.131 
[3] 
Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a twospecies chemotaxiscompetition system with two signals. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 36173636. doi: 10.3934/dcds.2018156 
[4] 
Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a twospecies chemotaxis model with nonlocal terms. Discrete & Continuous Dynamical Systems  B, 2017, 22 (10) : 38393874. doi: 10.3934/dcdsb.2017193 
[5] 
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems  S, 2021, 14 (3) : 11811195. doi: 10.3934/dcdss.2020226 
[6] 
Xu Pan, Liangchen Wang. On a quasilinear fully parabolic twospecies chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021047 
[7] 
Xie Li, Yilong Wang. Boundedness in a twospecies chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems  B, 2017, 22 (7) : 27172729. doi: 10.3934/dcdsb.2017132 
[8] 
Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a twospecies chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems  B, 2020, 25 (1) : 191221. doi: 10.3934/dcdsb.2019178 
[9] 
Chengxin Du, Changchun Liu. Time periodic solution to a twospecies chemotaxisStokes system with $ p $Laplacian diffusion. Communications on Pure & Applied Analysis, , () : . doi: 10.3934/cpaa.2021162 
[10] 
TaiChia Lin, ZhiAn Wang. Development of traveling waves in an interacting twospecies chemotaxis model. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 29072927. doi: 10.3934/dcds.2014.34.2907 
[11] 
Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a twospecies chemotaxis system. Discrete & Continuous Dynamical Systems  B, 2019, 24 (4) : 15691587. doi: 10.3934/dcdsb.2018220 
[12] 
Wenji Zhang, Pengcheng Niu. Asymptotics in a twospecies chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems  B, 2021, 26 (8) : 42814298. doi: 10.3934/dcdsb.2020288 
[13] 
KuangHui Lin, Yuan Lou, ChihWen Shih, TzeHung Tsai. Global dynamics for twospecies competition in patchy environment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 947970. doi: 10.3934/mbe.2014.11.947 
[14] 
Tobias Black. Global existence and asymptotic stability in a competitive twospecies chemotaxis system with two signals. Discrete & Continuous Dynamical Systems  B, 2017, 22 (4) : 12531272. doi: 10.3934/dcdsb.2017061 
[15] 
Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a twospecies chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems  B, 2020, 25 (12) : 45854601. doi: 10.3934/dcdsb.2020114 
[16] 
Youshan Tao, Michael Winkler. Boundedness vs.blowup in a twospecies chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems  B, 2015, 20 (9) : 31653183. doi: 10.3934/dcdsb.2015.20.3165 
[17] 
HaiYang Jin, Tian Xiang. Convergence rates of solutions for a twospecies chemotaxisNavierStokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems  B, 2019, 24 (4) : 19191942. doi: 10.3934/dcdsb.2018249 
[18] 
Masaaki Mizukami. Boundedness and asymptotic stability in a twospecies chemotaxiscompetition model with signaldependent sensitivity. Discrete & Continuous Dynamical Systems  B, 2017, 22 (6) : 23012319. doi: 10.3934/dcdsb.2017097 
[19] 
Masaaki Mizukami. Improvement of conditions for asymptotic stability in a twospecies chemotaxiscompetition model with signaldependent sensitivity. Discrete & Continuous Dynamical Systems  S, 2020, 13 (2) : 269278. doi: 10.3934/dcdss.2020015 
[20] 
Yan Li. Emergence of large densities and simultaneous blowup in a twospecies chemotaxis system with competitive kinetics. Discrete & Continuous Dynamical Systems  B, 2019, 24 (10) : 54615480. doi: 10.3934/dcdsb.2019066 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]