Ask Mr. Smarty Plants

Jerusalem thorn is an evergreen tree, growiing to 12 to 36 ft. tall, blooms yellow April to August, has medium water use, needs sun or part shade. It is subject to freeze damage if grown too far north, even as far as North Texas. 

This USDA Plant Profile shows the plant growing in some parts of South and Central Texas and the Big Bend area. It is native to the Southwestern United States and Mexico. It has escaped cultivation in Florida and California, both magnets for invasive plants, native or non-native, and it is regarded there as an invasive weed. 

in answer to your question. a native plant is just that, its status doesn't change, as long as it is growing in the area where it is native. So, yes, it's native to Central Texas. So far as we know, it has not become invasive in Central Texas. Native plants seldom do become invasive in their home territory, as they and the other native plants around them have all learned to live in the environment over eons of experience with the soil, climate, rainfall, etc. That environment has its own checks and balances, and human intervention, bringing in non-native plants or through urban development, frequently causes the development of invasive plants and, sometimes, the near-extinction of native plants. 

Parkinsonia aculeata

Parkinsonia aculeata

Parkinsonia aculeata

Parkinsonia aculeata