I have lived in various neighborhoods in Austin since 1971. I have seen my share of coyotes in town. When I lived off of Far West it was not infrequent that I would see one early in the morning while taking my son to band at Anderson.
Like opossums and raccoons, coyotes are very adaptable animals and like it or not have established themselves in the urban landscape. With the extensive networks of greenbelts, parks, golf courses and creek bottoms we have in Austin they have full access to all of Austin. I have friends along Shoal Creek, in NW Hills and in our adjoining neighborhoods that equally report sightings. And yes they will catch and eat your cat or small dog if given the opportunity.
That said, It would be both impossible and impractical to even attempt to eliminate them from the urban landscape. It would be just as effective as trying to get rid of the urban deer population. In my opinion it just ain't gonna happen.
Even if those spotted around Mueller could be captured/killed it would only create a vacuum that soon would be replaced.
It would be more effective to not provide them the opportunity for an easy meal.
My point exactly: "It would be more effective to not provide them the opportunity for an easy meal." If they were eliminated, what they would normally eat would be the next problem for everyone. For now, loose pets and even barking dogs in back yards are like turning on the blue light special for these creatures. For my part, I carry protection in case I am threatened; otherwise, I wouldn't mess with any of it, other than to suggest making it more attractive for them to find a meal elsewhere.
Whether or not they should be relocated, they are losing their fear of humans, which is a really bad sign. I did a bit of research on coyotes in urban areas and there are techniques called "coyote hazing" that help keep coyotes from being habituated to humans and hopefully make them less dangerous. If everyone in the neighborhood uses these methods whenever spotting a coyote, it may help protect people and their pets. Here's a flyer on coyote hazing for reference: http://www.humanesociety.org/assets/pdfs/wild_neighbors/coyote_hazing.pdf
Coyotes are often the physical manifestation of a soul, that for whatever reason, has become lost and yearns for the warm embrace of spiritual redemption. Some of these poor creatures take the steps required to find their salvation and travel onto the next plane of spiritual transformation.
However, most recoil from that challenging path and instead quest in the opposite direction, wreaking havoc, murdering indescriminately, and preying upon the weak and helpless.
The best strategy for dealing with the steady encroachment of Coyoté is to pray for their souls and hang a wreath of rosemary on your door.
When I lived in Elgin before moving to Mueller, I was confronted with a clan of Coyoté. Eventually I was forced to move. Sadly, many who gave me advice during that difficult time, told me that evacuation is often the only recourse for the stricken. I have not yet seen one of the Coyoté reported in this thread, but it troubles me to think that I may be forced to move again on account of these loathsome beasts.