# American Institute of Mathematical Sciences

August  2018, 11(4): 891-909. doi: 10.3934/krm.2018035

## Traveling wave and aggregation in a flux-limited Keller-Segel model

 1 Institut Camille Jordan, UMR 5208 CNRS/Université Claude Bernard Lyon 1, and Project-team Inria NUMED, Lyon, France 2 Sorbonne Universités, UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions UMR CNRS 7598, Université Paris Diderot, Inria de Paris, F75005 Paris, France 3 Graduate School of Simulation Studies, University of Hyogo, Kobe 650-0047, Japan

Received  September 2017 Revised  January 2018 Published  April 2018

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term.

Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response.

In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counter-balancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.

Citation: Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035
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##### References:
The schematic of domain decomposition in the relative coordinate system $\xi$ introduced in Sec. 2.3
Figure (a) shows the solution curves of Eq. (44) in $\hat \xi_c$-$\hat \chi$ plane with variation in the diffusion constant $d$, while the proliferation rate $p = 0.5$ is fixed. Figure (b) shows the parameter regime which satisfies the constraints Eqs. (39) and (40). Here the modulation amplitude $\hat \chi = 3.0$ is fixed. The contour shows the peak value of population wave, i.e., $\alpha$ defined in Eq. (34). The symbols "$\times$" in figure (b) show the solutions of Eq. (44) with $\hat \chi$ = 3.0 and $d =$1, 5, 10, 25, and 50, respectively, from left to right
, the squares (Type Ⅲ) refer to Fig. 6, the triangles (Type Ⅳ) refer to Fig. 7, and the diamonds (Type Ⅴ) refer to Fig. 8. The colors of each symbol show the maximum value of population density in the spatial profile. The diffusion coefficient $d$ and proliferation rate $p$ are fixed as $d = 4$ and $p = 0.5$. The dotted horizontal line shows the critical value determined by the instability condition">Figure 3.  The diagram of different types of numerical solutions with variation in the modulation $\chi$ and stiffness $\delta^{-1}$. The circles (Type Ⅰ and Ⅱ) refer to Fig. 5, the squares (Type Ⅲ) refer to Fig. 6, the triangles (Type Ⅳ) refer to Fig. 7, and the diamonds (Type Ⅴ) refer to Fig. 8. The colors of each symbol show the maximum value of population density in the spatial profile. The diffusion coefficient $d$ and proliferation rate $p$ are fixed as $d = 4$ and $p = 0.5$. The dotted horizontal line shows the critical value determined by the instability condition
. The vertical line on the horizontal axis indicates the critical value of the instability condition. The dotted lines show the minimum speeds obtained by Eq. (16). The numbers (Ⅰ)-(Ⅴ) illustrate the solution types shown in Fig. 3">Figure 4.  The speed of propagating front with variation in the modulation $\chi$ and stiffness $\delta^{-1}$. The values of parameters $d$ and $p$ are same as in Fig. 3. The vertical line on the horizontal axis indicates the critical value of the instability condition. The dotted lines show the minimum speeds obtained by Eq. (16). The numbers (Ⅰ)-(Ⅴ) illustrate the solution types shown in Fig. 3
The snapshots of monotonic and non-monotonic traveling waves for $2\chi/(\pi\delta) = 0.01$ (a) and $2\chi/(\pi\delta) = 5.0$ (b), respectively. The other parameters are set as $\hat \chi = 1.5$, $d = 4.0$, and $p = 0.5$
The snapshots of backward traveling wave for $\tilde \chi = 3.0$, $2\chi/(\pi\delta) = 9$, $d = 4.0$, and $p = 0.5$
The snapshots of periodic pattern formation with a moving front in forward (a) and backward (b) directions. The modulation parameter is set as $\hat\chi = 1.0$ (a) and $\hat\chi = 3.0$ (b). The other parameters are set as $2\chi/(\pi\delta) = 10.0$, $d = 4.0$, and $p = 0.5$ in both (a) and (b)
The snapshots of pattern formation of localized spikes. The parameters are set as $\hat \chi = 3.0$, $2\chi/(\pi\delta) = 20.0$, $d = 4.0$, and $p = 0.5$. Note that the results at $t =$200 and 500 are almost overlapped only except the region around $\hat x = 350$
The comparison of the traveling speed measured from numerical solution $c^*$ to the minimum traveling speed $c_\mathrm{min}$ obtained by Eq. (16). The modulation amplitude $\hat\chi = 1.5$ and diffusion constant $d = 1.0$ are fixed while the proliferation rate $p$ varies. The dispersion relation Eq. (14) between the traveling speed and exponential decay of population density far ahead the front, i.e., $c(\lambda^*)$ is also plotted. The way to measure $c^*$ and $\lambda^*$ is described in the last paragraph of Sec. 3.1. Note that $c^*$ and $c(\lambda^*)$ are almost overlapped in the figure
(i.e., Type Ⅲ in Fig. 4) at time $\hat t = 500$">Figure 10.  The snapshot of chemotactic drift speed $U_\delta$ for Fig. 6 (i.e., Type Ⅲ in Fig. 4) at time $\hat t = 500$
Numerical solutions for large-stiffness parameters are compared to the analytical solution for the stiff flux Eq. (18), i.e., $2\chi/(\pi\delta)\rightarrow \infty$. The modulation amplitude $\chi$ and proliferation rate $p$ are fixed as $\hat\chi = 2.5$ and $p = 0.5$, respectively. The diffusion coefficient $d$ is set as $d = 4$ in figure (a) and $d = 16$ in figure (b)
The accuracy tests performed with different numbers of mesh interval $I$, i.e., $I$ = 5000, 10000, 20000. The subscripts $f$ and $c$ represent the finer and coarser mesh systems, respectively. The traveling speeds $c^*$ and exponential decay $\lambda^*$ are directly measured from the numerical solutions
 $I_f$ -$I_c$ $|\frac{c^*_f-c^*_c}{c^*_f} |$ $|\frac{\lambda^*_f-\lambda^*_c}{\lambda^*_f}|$ $\frac{c^*_f-c(\lambda_f^*)}{c(\lambda_f^*)}$ 10000 -5000 $1.7\times 10^{-3}$ $2.8\times 10^{-3}$ $1.1\times 10^{-3}$ 20000 -10000 $3.6\times 10^{-4}$ $5.6\times 10^{-4}$ $8.1\times 10^{-4}$
 $I_f$ -$I_c$ $|\frac{c^*_f-c^*_c}{c^*_f} |$ $|\frac{\lambda^*_f-\lambda^*_c}{\lambda^*_f}|$ $\frac{c^*_f-c(\lambda_f^*)}{c(\lambda_f^*)}$ 10000 -5000 $1.7\times 10^{-3}$ $2.8\times 10^{-3}$ $1.1\times 10^{-3}$ 20000 -10000 $3.6\times 10^{-4}$ $5.6\times 10^{-4}$ $8.1\times 10^{-4}$
The decay rate $\lambda$ defined in Eq. (12) and the distance from the peak of chemoattractant to the position where the population density equals to $\rho_c$, i.e., $\xi_c = x_c-x_S$ where $\rho(x_c) = \rho_c$ and $\partial_x S(x_S) = 0$, with variation in the stiffness. The modulation amplitude $\chi$ and proliferation rate $p$ are fixed as $\hat \chi = 2.5$ and $p = 0.5$, respectively
 $d=4.0$ $d=16.0$ $\frac{2\chi}{\pi\delta}$ $\lambda/\sqrt{p}$ $\hat \xi_c$ $\frac{2\chi}{\pi\delta}$ $\lambda/\sqrt{p}$ $\hat \xi_c$ 7.0 1.30 3.08 21.0 1.065 5.57 8.0 1.26 2.96 23.0 1.062 5.44 9.0 1.23 2.79 25.0 1.059 5.28 10.0 1.20 2.61 20.0 1.058 5.16 $\infty$ 1.00 3.09 $\infty$ 1.00 6.95
 $d=4.0$ $d=16.0$ $\frac{2\chi}{\pi\delta}$ $\lambda/\sqrt{p}$ $\hat \xi_c$ $\frac{2\chi}{\pi\delta}$ $\lambda/\sqrt{p}$ $\hat \xi_c$ 7.0 1.30 3.08 21.0 1.065 5.57 8.0 1.26 2.96 23.0 1.062 5.44 9.0 1.23 2.79 25.0 1.059 5.28 10.0 1.20 2.61 20.0 1.058 5.16 $\infty$ 1.00 3.09 $\infty$ 1.00 6.95
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