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Abstract. Consider a Banach lattice E and two positive operators S,T: E -> E that satisfy 0 *c S =c T. In [2,3] we examined the case when T is a compact (or weakly compact) operator and studied what effect this had on an operator (such as S) dominated by T. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator T, to ensure that S (or some power of S) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters.Theorem. Let E be a Banach lattice, and let S,T: E -» E be two positive operators such that 0 < S =c T. If T is compact then(1) S3 is a compact operator (although S2 need not be compact); (2) S2 is a Dunford-Pettis and weakly compact operator ( although S need not be); (3) S is a weak Dunford-Pettis operator.In another direction, our techniques and results will be related to the lattice stmcture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.