  We consider a nonlocal generalization of the Fisher-KPP equation in one
spatial dimension. As a parameter is varied the system undergoes a Turing
bifurcation. We study the dynamics near this Turing bifurcation. Our results
are two-fold. First, we prove the existence of a two-parameter family of
bifurcating stationary periodic solutions and derive a rigorous asymptotic
approximation of these solutions. We also study the spectral stability of the
bifurcating stationary periodic solutions with respect to almost co-periodic
perturbations. Secondly, we restrict to a specific class of exponential kernels
for which the nonlocal problem is transformed into a higher order partial
differential equation. In this context, we prove the existence of modulated
traveling fronts near the Turing bifurcation that describe the invasion of the
Turing unstable homogeneous state by the periodic pattern established in the
first part. Both results rely on a center manifold reduction to a finite
dimensional ordinary differential equation.
