Figure 1

Simulation of island patterns in 1D chemotaxis system (1), starting from a perturbed initial steady state $z(x,0)=z_0(x)=1.0\pm 0.1\times \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}$, $c(x,0)=c_0(x)\equiv 0$ with computer generated random number $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\in (0,1)$, $L=10$, $D=0.33$, $\alpha =80$, and von Neumann boundary conditions $z_x(0,t)=z_x(L,t)=c_x(0,t)=c_x(L,t)=0$.Reuse & Permissions

Figure 2

Simulation of wave propagation in 1D chemotaxis system (1), starting from a singular perturbation $z(0,0)=z_0(0)=1.2$, $z(x,0)=z_0(x)=1.0$ for $x\ne 0$, $c(x,0)=c_0(x)\equiv 0$ of the initial steady state $z(x,0)=z_0(x)\equiv 1.0$, $c(x,0)\equiv 0$, $L=10$, $D=0.33$, $\alpha =80$, and von Neumann boundary conditions $z_x(0,t)=z_x(L,t)=c_x(0,t)=c_x(L,t)=0$.Reuse & Permissions

Figure 3

Enforcement of specific cell distributions according to (2), starting from a randomly perturbed steady state as in Fig. 1, $L=1$, $T$ fixed, $D=0.33$, $\alpha =80$, von Neumann boundary conditions $z_x(0,t)=z_x(L,t)=c_x(0,t)=0$. A linear model $c_x(L,t)=c(L,t)-u_0(t)$ is used for the boundary flux control at $x=L$, $u_0(t)$ is the controlled external chemoattractant concentration. Left: Real distribution $z(x,T)$ at fixed time $T=1$ (top) and $T=2$ (bottom) arising from controlled system dynamics and desired cell distributions $z_T(x)=1.5\times \exp [-10(x-0.5{)}^2]$ (top) and $z_T(x)=1.5(1-x^2)$ (bottom). Right: Corresponding optimal control functions $u_0(t)$ according to problem (2) with $\lambda =0$ computed with the optimal control package MUSCOD-II.Reuse & Permissions

Figure 4

Top: Inhibition of pattern formation according to (2), starting from a randomly perturbed steady state as in Fig. 1, $L=1$, $T=2$ fixed, $D=0.33$, $\alpha =80$, $\lambda =0$, von Neumann boundary conditions $z_x(0,t)=z_x(L,t)=c_x(0,t)=0$. A linear model $c_x(L,t)=c(L,t)-u_0(t)$ is used for the boundary flux control at $x=L$, $u_0(t)$ is the controlled external chemoattractant concentration. Bottom: Inhibition of wave propagation, starting from a singularly perturbed initial steady state (arrow indicates the perturbation at $x=0$) as in Fig. 2, $L=1$, $T$ free and subject to minimization. Left: Real distributions $z(x,T)$ at fixed time $T=2$ (top) and optimal time $T=3.7$ (bottom) arising from controlled system dynamics and desired homogeneous cell distributions $z_T(x)\equiv 1.0$. Right: Corresponding optimal control functions $u_0(t)$ according to control problem (2) with $\lambda =0$ (top) and $\lambda =0.001$ (bottom) computed with the optimal control package MUSCOD-II.Reuse & Permissions