Spreading speeds of nonlocal KPP equations in heterogeneous media

In this paper, we prove the existence of the spreading speed of nonlocal KPP equations in two cases: 1. The media is almost periodic and the kernel of diffusion is continuous; 2. The media is periodic and the diffusion is not continuous but weak irreducible. To do this, we develop the theory of generalized principal eigenvalues, the method homogenization and special Harnack's inequalities of nonlocal diffusion equations.