# American Institute of Mathematical Sciences

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 and 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. [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. [3] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933. [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. [5] M. Campos-Pinto, J. A. Carrillo, F. Charles and Y. -P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. [6] J. Carrillo, M. Di Francesco, A. Figalli, T. 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. [7] J. A. Carrillo, A. 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. [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), 304-338. [9] K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717. [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. [11] M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808. [12] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595-615. [13] C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. [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), 359-380. [15] L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 987-1015. [16] D. Helbing, W. 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. [17] D. D Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Physical Review Letters, 95 (2005), 226106. [18] F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127. [19] F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. [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. [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), 1590-1592. [22] A. Mackey, T. Kolokolnikov and A. Bertozzi, Two-species particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1411-1436. [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), 13259-13263. [24] H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. [25] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. [26] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), Springer-Verlag, New York, 1998, Applications. [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. [28] K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. [29] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. [30] C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. [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.

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. [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. [3] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933. [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. [5] M. Campos-Pinto, J. A. Carrillo, F. Charles and Y. -P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. [6] J. Carrillo, M. Di Francesco, A. Figalli, T. 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. [7] J. A. Carrillo, A. 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. [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), 304-338. [9] K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 1681-1717. [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. [11] M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808. [12] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595-615. [13] C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. [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), 359-380. [15] L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 987-1015. [16] D. Helbing, W. 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. [17] D. D Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Physical Review Letters, 95 (2005), 226106. [18] F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127. [19] F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. [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. [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), 1590-1592. [22] A. Mackey, T. Kolokolnikov and A. Bertozzi, Two-species particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1411-1436. [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), 13259-13263. [24] H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. [25] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. [26] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), Springer-Verlag, New York, 1998, Applications. [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. [28] K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. [29] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. [30] C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. [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.
Example 1. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot). The evolution shows the synchronising dynamics after first collision
Example 2. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot). The evolution shows the non-synchronising dynamics after first collision
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
Example 4. Snapshots of $\rho_{1}$ (red solid line) and $\rho_{2}$ (blue dashdot)
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