Welcome to the forum @mkretzschmar 
Are the axes perhaps labelled the wrong way? If we assume that the x-axis represents age and the y-axis time, then I would say we see exactly what you describe:
- a plateau at age 20-30 at t=0 which is advected
- a constant inflow from the “left” (age 0) by fixing the value there
The confusion might have been that in the above code examples I used the arguments in the order S(a,t), but you used S(t,a), etc.