I currently need to solve numerically the following reaction-diffusion equation:
$$\partial_tu=\partial^2_xu+u-u^2$$
For this purpose, I use the following numerical scheme (Crank-Nicolson??): $$ \frac{u(x,t+\delta t)-u(x,t)}{\delta t} = \frac{1}{2(\delta x)^2} \left[u(x+\delta x,t+\delta t)-2u(x,t+\delta t)+u(x-\delta x,t+\delta t)+u(x+\delta x,t)-2u(x,t)+u(x-\delta x,t)\right] + u(x,t)-u^2(x,t)$$
How can I investigate the stability of this scheme?