Isik and Turkoglu [Some Fixed Point Theorems in Ordered Partial Metric Spaces, Journal of Inequalities and Special Functions, Volume 4 Issue 2(2013), Pages 13-18] established new fixed point results in complete partial metric spaces. The purpose of this paper is to introduce a dual contraction of rational type and to obtain some new fixed point results in dual partial metric spaces for dominating and dominated mappings. These results extend various comparable results existing in literature. Moreover, we give an example that shows the usefulness and effectiveness of these results among corresponding fixed point theorems established in partial metric spaces.