# 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 & Continuous Dynamical Systems - B, 2017, 22 (6) : 2121-2146. doi: 10.3934/dcdsb.2017088
##### References:

show all references

##### References:
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)
 [1] Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a two-species kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359-380. doi: 10.3934/krm.2015.8.359 [2] Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131 [3] Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156 [4] Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. 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, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020226 [6] Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132 [7] Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178 [8] Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907 [9] Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220 [10] Kuang-Hui Lin, Yuan Lou, Chih-Wen Shih, Tze-Hung Tsai. Global dynamics for two-species competition in patchy environment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 947-970. doi: 10.3934/mbe.2014.11.947 [11] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [12] Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165 [13] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [14] Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094 [15] Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249 [16] Yan Li. Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5461-5480. doi: 10.3934/dcdsb.2019066 [17] Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015 [18] Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067 [19] Salvatore Rionero. A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences & Engineering, 2006, 3 (1) : 189-204. doi: 10.3934/mbe.2006.3.189 [20] Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195

2018 Impact Factor: 1.008