Ask Mr. Smarty Plants

This is a redo of two previous answers to this question. Someone better at plant identification than this member of the Mr. Smarty Plants team has suggested that it is more likely that the plant in question is Nolina texana (Texas sacahuista). One of its common names is "Texas beargrass." It is neither a grass nor a succulent but a member of the lily family. This USDA Plant Profile map shows that it is, indeed, native to the area in question.In fact, it is grown on the grounds of the Lady Bird Johnson Wildflower Center.

However, our original answer to the question is still the same. It is spreading out because, while it can survive drought and heat, it still loves more moisture and cultivated soil. It is a lovely plant and we certainly would not recommend trying to remove it, but just control. The open spaces in the first plant probably indicate the aging of the plant, so it is naturally trying to propagate, as the primary goal for every living thing is to make more of itself.

We certainly don't recommend herbicides; they can have a negative effect on your soils and on the wildflowers you are trying to grow. Sorry it took us so long to correctly identify the plant in question.