The random walk of a particle on an asymmetric chain in the presence of an attractive center, possibly trapping, is examined by means of the equivalent transfer rates technique. Both the situations of ordered and disordered hopping rates are studied. It is assumed that initially the particle is located on the attractive center. The (average) probability of presence of the particle at its initial point is computed as a function of time. In the ordered case this quantity decreases exponentially toward its limiting value (with in certain cases an inverse power-law prefactor), while in the presence of disorder it decreases according to a power law, with an exponent depending both on disorder and on asymmetry. When the possibility of trapping is taken into account, this model is relevant for the description of the transfer of energy in a photosynthetic system. The amount of energy conserved within the chain, as a function of time, and the average lifetime of the particle before it is captured by the trap are examined in both ordered and disordered situations.