Motivated by various disordered propagation problems with competing channels, I study the representative problem of Anderson localization on an asymmetric two-leg ladder.  The problem is solved by the Fokker-Planck approach, which is exact in the weak disorder limit.  The localization radius of various one dimensional systems, such as a polaritons or other hybrid particles, can be investigated by this model.  These applications correspond to parametrically different intra-chain hopping integrals and/or different disorder amplitudes on the two legs, situations in which it is non-trivial to predict what dominates the transport in the joint system.
An extended Dorokhov-Mello-Pereya-Kumar (DMPK) equation is obtained and solved analytically.  Two localization lengths are obtained as functions of the parameters of the model.  We find that: 1) Near the resonance energy (where the dispersion curves of the two decoupled and  disorder-free
chains intersect) the "slow'' chain dominates the   localization properties of the ladder. 2) Away from the resonance the   "fast'' chain dominates the transmission probability.