I suppose it depends which particular systems one is speaking about. A zoo or a botanical garden are artificial but they may have greater biodiversity than some natural systems, by design. A zoo is a sort of artificial biodiversity reserve, so that's to be expected.
But generally speaking natural systems tend to hold a greater biodiversity than artificial ones, because most of the latter are agricultural. Crop fields, pastures, or woodlots are generally the result of man (the farmer) favoring one species or a small number of "useful" species at the expense of all the others. The species that is being cropped is given a very strong advantage. Eg in a corn field the farmer will try to weed out all non-corn plants (called "weeds") and will try and kill all animal species feeding off the corn (called "pests"). This results in less biodiversity in a "clean" corn field than in whatever would grow on it naturally.