  We obtain almost optimal differential Harnack inequalities for a class of
nonlinear parabolic equations on Riemannian manifolds with Bakry-\'{E}mery
Ricci curvature bounded below, which includes the classical Fisher-KPP equation
and Newell-Whitehead equation. Compared to existing research, we do not impose
any additional conditions on the positive solutions. As its application, we
derive some optimal Liouville properties.
