  This paper is concerned with asymptotic persistence, extinction and spreading
properties for non-cooperative Fisher-KPP systems with space-time periodic
coefficients. Results are formulated in terms of a family of generalized
principal eigenvalues associated with the linearized problem. When the maximal
generalized principal eigenvalue is negative, all solutions to the Cauchy
problem become locally uniformly positive in long-time, at least one space-time
periodic uniformly positive entire solution exists, and solutions with
compactly supported initial condition asymptotically spread in space at a speed
given by a Freidlin-G{\"a}rtner-type formula. When another, possibly smaller,
generalized principal eigenvalue is nonnegative, then on the contrary all
solutions to the Cauchy problem vanish uniformly and the zero solution is the
unique space-time periodic nonnegative entire solution. When the twogeneralized
principal eigenvalues differ and zero is in between, the long-time behavior
depends on the decay at infinity of the initial condition. The proofs rely upon
double-sided controls by solutions of cooperative systems. The control from
below is new for such systems and makes it possible to shorten the proofs and
extend the generality of the system simultaneously.
