Analytical calculations of extragalactic cosmic ray spectra above $\sim {10}^{17}\mathrm{eV}$ are often performed assuming continuous source distributions, giving rise to spectra that depend little on the propagation mode, be it rectilinear or diffusive. We perform trajectory simulations for proton primaries in the probably more realistic case of discrete sources with a density of $\sim {10}^{-5}{\mathrm{Mpc}}^{-3}$. We find two considerable nonuniversal effects that depend on source distributions and magnetic fields: First, the primary extragalactic cosmic ray flux can become strongly suppressed below a few ${10}^{18}\mathrm{eV}$ due to partial confinement in magnetic fields surrounding sources. Second, the secondary photon to primary cosmic ray flux ratio between $\simeq 3\times {10}^{18}\mathrm{eV}$ and $\simeq {10}^{20}\mathrm{eV}$ decreases with decreasing source density and increasing magnetization. As a consequence, in acceleration scenarios for the origin of highest energy cosmic rays the fraction of secondary photons may be difficult to detect even for experiments such as Pierre Auger. The cosmogenic neutrino flux does not significantly depend on source density and magnetization.

• Received 15 March 2007

DOI:https://doi.org/10.1103/PhysRevD.75.103001