Stability and bifurcation diagram for a one-dimensional shadow reaction-diffusion system of substrate depletion type

We are concerned with a shadow reaction-diffusion system of substrate depletion type in a finite interval$\begin{align} \begin{cases} \partial_tA = {\varepsilon}^2A_{xx}+f(\xi)A^3-A & \text{for}\ 0<x<1,\ 0<t<T,\\ \tau\partial_t\xi = \gamma-f(\xi)\int_0^1A^3dx-\xi & \text{for}\ 0<t<T,\\ A_x(0,t) = A_x(1,t) = 0 & \text{for}\ 0<t<T, \end{cases} \end{align}$where $ A(x,t) $ and $ \xi(t) $ are nonnegative unknown functions, and $ f\in C^1[0,\infty) $ is a nonnegative increasing function satisfying certain conditions. We determine the complete bifurcation diagram of the positive stationary solutions if $ \gamma>0 $ is larger than a certain value. We also study the stability of every positive stationary solution. Specifically, we show that some $ 1 $-mode solutions are stable for small $ \tau>0 $ and large $ \gamma>0 $, and that $ n $-mode solutions, $ n\ge 2 $, are unstable for all $ \tau>0 $. In particular, the case where $ f(\xi) = \xi^2 $ is studied in detail.