  The global dynamics of the two-species Lotka-Volterra competition patch model
with asymmetric dispersal is classified under the assumptions of weak
competition and the weighted digraph of the connection matrix is strongly
connected and cycle-balanced. It is shown that in the long time, either the
competition exclusion holds that one species becomes extinct, or the two
species reach a coexistence equilibrium, and the outcome of the competition is
determined by the strength of the inter-specific competition and the dispersal
rates. Our main techniques in the proofs follow the theory of monotone
dynamical system and a graph-theoretic approach based on the Tree-Cycle
identity.
