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

Figure 1.  The schematic of domain decomposition in the relative coordinate system $\xi$ introduced in Sec. 2.3

Figure 2.  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

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

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

Figure 5.  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 $p = 0.5$

Figure 6.  The snapshots of backward traveling wave for $\tilde \chi = 3.0$, $2\chi/(\pi\delta) = 9$, $d = 4.0$, and $p = 0.5$

Figure 7.  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)

Figure 8.  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 $\hat x = 350$

Figure 9.  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

Figure 10.  The snapshot of chemotactic drift speed $U_\delta$ for Fig. 6 (i.e., Type Ⅲ in Fig. 4) at time $\hat t = 500$

Figure 11.  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)