Welcome to Arizona where Open Range is the norm. Those are the ranchers’ cows. If you don't want them on your property, fence them out. If you don't want them in your gardens, swimming pools, etc. then fence them out with a PROPER fence as per either the State Trust fencing standards or the Public Land standard depending on what type of available range land is nearest to you.
Generally speaking if you put steel posts every 18 feet and metal stays at least every 7 feet between those, you have a good start… then they say the bottom barb wire should be no more than 10 inches above the ground, then 12 inches between each of the next three barb wire strands. Making a total of 4 rows of barbed wire with the top wire at approximately 46 inches tall. Ranchers who abut state highways swear they have had excellent results with 5 wire fencing with the top wire at 54 inches high and the bottom at 8" - but dang are those fences hard to crawl over or under!
If your dogs chase these cattle and the rancher catches them, he will likely shoot your dogs – and like it or not he will be in the right to do so. We are lucky to have a bit of a soft hearted rancher, he will warn you/educate you once. We do have a neighbor who refuses to fence his acreage, and his dogs are very well trained cattle dogs, they chase the ranchers’ cows exactly to the property line and then they stop – apparently this is an acceptable method for this particular rancher, because the dogs are all still alive!
Those range cows have water for them within a 3-4 mile journey – if you know what’s good for you, you will heed my warning – DO NOT PUT WATER OUT FOR THEM!!!!! Once you do this – they will always come back and when the cool water trough that you filled becomes empty cause Bessie brought 4 of her friends for a drink then they will bust into your flimsy dog run fence and drink the dog’s water, and the cats water, and the goat’s water…. Do NOT put water out for the cute range cows. If they are thirsty – they know where to go!!!!!
You ask how I know this? Well...I put water out for #187 (she looks like a couple year old cow), and she came back night after night, she would drink the bucket dry and make the dogs bark, I would come outside and fill the bucket again and she would come back up the driveway and empty it again – she had me so trained!!!! I thought it was cute, because several years in a row she had her calf on my property, she felt safe here. And her visits and requests for water usually only lasted 4 or 5 days. But then one year she brought her friends, FIVE cows demanding I fill the bucket over and over again –I shooed them down the driveway towards their water tank at the bottom of the hill. Lesson learned – do NOT feed or water the ranchers’ cattle. They need to be controlled by the water, if the rancher needs to move them he can shut off a water source and push them to another, then they will stay closer to the other water hole. This is open-range cattle management.
Cattle in the southwest do not look like the round fat Black Angus you find in the belly deep green grass fields in Colorado. The breeds here will look thin even when they are fit. They can range on scrub, weeds and poor grass and still gain weight. If the cattle were severely underweight there would be no babies as unfit cattle do not get pregnant as their bodies know they could not support another life. That makes them smarter than some humans I know.
So fence’em out, do not feed or water them, meet your rancher and enjoy the scenery, and remember, "Life Begins Where the Pavement Ends."