Problem #$1$: In this problem, you will work on a $1$D$,3$ cell Turing model, with zero boundary conditions.

Let the activation radius and weight be set at $R_A=1$ and $w_A=1,$ respectively. Let the inhibition radius

and weight be set at $R_{-}I=3$ and $w_{-}I=k,$ respectively. Here $\frac{?}{\text{b}\text{a}\text{r}}(k)$ is a parameter. For initial state $S_0=1,1,0,$

determine general conditions on $k($if any are possible$)$ so that $S_n=1,1,0$ for all $n=0,1,2,$dots

Problem #$2$: Using the same Turing model as in Problem #$1,$ but now with initial state S$\_0=1,1,1,$

determine general conditions on $k($if any are possible$)$ so that $S_{-}n=1,1,1$ for all $n=0\frac{,}{b}ar(1),$dots

Problem #$3$: Using the same Turing model as in Problem #$2,$ again with initial state S$\_0=1,1,1,$ determine

general conditions on $k($if any are possible$)$ so that $S_1{S}_0$ and $S_2=S_0.$