  In this paper, we investigate propagation phenomena for KPP bulk-surface
systems in a cylindrical domain with general section and heterogeneous
coefficients. As for the scalar KPP equation, we show that the asymptotic
spreading speed of solutions can be computed in terms of the principal
eigenvalues of a family of self-adjoint elliptic operators. Using this
characterization, we analyze the dependence of the spreading speed on various
parameters, including diffusion rates and the size and shape of the section of
the domain. In particular, we provide new theoretical results on several
asymptotic regimes like small and high diffusion rates and sections with small
and large sizes. These results generalize earlier ones which were available in
the radial homogeneous case. Finally, we numerically investigate the issue of
shape optimization of the spreading speed. By computing its shape derivative,
we observe, in the case of homogeneous coefficients, that a disk either
maximizes or minimizes the speed, depending on the parameters of the problem,
both with or without constraints. We also show the results of numerical shape
optimization with non homogeneous coefficients, when the disk is no longer an
optimizer.
