August  2017, 22(6): 2121-2146. doi: 10.3934/dcdsb.2017088

Synchronising and non-synchronising dynamics for a two-species aggregation model

1. 

Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

2. 

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

3. 

INRIA-Paris-Rocquencourt, 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 Jean-Baptiste Clément, 93430 Villetaneuse, France

Received  June 2015 Revised  January 2017 Published  March 2017

This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a two-species population in interaction appearing for instance in bacterial chemotaxis. Blow-up of classical solutions occurs in finite time. This raises the question to define measure-valued 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 measure-valued 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 measure-valued 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 blow-up 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.

Citation: Casimir Emako-Kazianou, Jie Liao, Nicolas Vauchelet. Synchronising and non-synchronising dynamics for a two-species aggregation model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2121-2146. doi: 10.3934/dcdsb.2017088
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, Finite-time blow-up of L-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. Google Scholar

[3]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933. 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), 2173-2189. Google Scholar

[5]

M. Campos-Pinto, J. A. Carrillo, F. Charles and Y. -P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted.Google Scholar

[6]

J. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. Google Scholar

[7]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. Google Scholar

[8]

J. A. CarrilloF. JamesF. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338. Google Scholar

[9]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717. Google Scholar

[10]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523-537. Google Scholar

[11]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808. Google Scholar

[12]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595-615. Google Scholar

[13]

C. EmakoC. GayrardA. BuguinL. N. de Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar

[14]

C. EmakoL. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359-380. Google Scholar

[15]

L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 987-1015. Google Scholar

[16]

D. HelbingW. Yu and H. Rauhut, Self-organization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177-208. Google Scholar

[17]

D. D Holm and V. Putkaradze, Aggregation of finite-size 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), 101-127. Google Scholar

[19]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. 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), 1355-1382. Google Scholar

[21]

T. LiuM. L. K. LangstonD. LiJ. M. PiggaC. PichonA. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 1590-1592. Google Scholar

[22]

A. MackeyT. Kolokolnikov and A. Bertozzi, Two-species particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1411-1436. Google Scholar

[23]

N. MittalE. O. BudreneM. P.r Brenne and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 13259-13263. Google Scholar

[24]

H. OthmerS. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. Google Scholar

[25]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. Google Scholar

[26]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), Springer-Verlag, New York, 1998, Applications.Google Scholar

[27]

J. SaragostiV. CalvezN. BournaveasA. BuguinP. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar

[28]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. 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, Springer-Verlag, Berlin, 2009.Google Scholar

[31]

J. H. Von BrechtD. UminskyT. 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, Finite-time blow-up of L-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. Google Scholar

[3]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933. 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), 2173-2189. Google Scholar

[5]

M. Campos-Pinto, J. A. Carrillo, F. Charles and Y. -P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted.Google Scholar

[6]

J. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. Google Scholar

[7]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. Google Scholar

[8]

J. A. CarrilloF. JamesF. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338. Google Scholar

[9]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717. Google Scholar

[10]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523-537. Google Scholar

[11]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808. Google Scholar

[12]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595-615. Google Scholar

[13]

C. EmakoC. GayrardA. BuguinL. N. de Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar

[14]

C. EmakoL. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359-380. Google Scholar

[15]

L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 987-1015. Google Scholar

[16]

D. HelbingW. Yu and H. Rauhut, Self-organization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177-208. Google Scholar

[17]

D. D Holm and V. Putkaradze, Aggregation of finite-size 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), 101-127. Google Scholar

[19]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. 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), 1355-1382. Google Scholar

[21]

T. LiuM. L. K. LangstonD. LiJ. M. PiggaC. PichonA. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 1590-1592. Google Scholar

[22]

A. MackeyT. Kolokolnikov and A. Bertozzi, Two-species particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1411-1436. Google Scholar

[23]

N. MittalE. O. BudreneM. P.r Brenne and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 13259-13263. Google Scholar

[24]

H. OthmerS. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. Google Scholar

[25]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. Google Scholar

[26]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), Springer-Verlag, New York, 1998, Applications.Google Scholar

[27]

J. SaragostiV. CalvezN. BournaveasA. BuguinP. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar

[28]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. 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, Springer-Verlag, Berlin, 2009.Google Scholar

[31]

J. H. Von BrechtD. UminskyT. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31pp. Google Scholar

Figure 1.  Example 1. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot). The evolution shows the synchronising dynamics after first collision
Figure 2.  Example 2. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot). The evolution shows the non-synchronising dynamics after first collision
Figure 3.  Example 3. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot). From time $t_0 = 0$ to $t_1\approx0.47$, $\mu_1$ moves toward $\nu_1$. From time $t_1\approx0.47$ to $t_2 \approx 1.04$, $\mu_1$ and $\nu_1$ travel together. The synchronising type changed at $t_2 $. After time $t_2$, $\mu_1$ overtakes $\nu_1$ and collapse with $\mu_2$ at $t_3\approx2.037$, and finally all the aggregates collapse at $t_4 \approx 2.32$. The evolution shows the transition from synchronising to non-synchronising dynamics
Figure 4.  Example 4. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot)
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