  In this manuscript, we study the positive solutions of the Finslerian
Fisher-KPP equation
  $$
  u_t=\Delta^{\nabla u} u+cu(1-u).
  $$
  The Fisher-KPP equation is widely applied and connected to many mathematical
branches. We establish the global gradient estimates on compact Finsler metric
measure manifold with the traditional $CD(K,N)$ condition, which is developed
by S. Ohta and K.-T. Sturm in Finsler geometry. Furthermore, With the
assistance of a new comparison theorem developed by the first author, we also
give the gradient estimate on forward complete noncompact locally finite
misalignment Finsler metric measure spaces with the mixed weighted curvature
bounded below and some non-Riemannian curvatures norm-bounded.
