Most apex predators are hunted out of existence to protect the marginal amounts of cattle they kill. See the great wolf slaughter happening in Montana currently. After they kill off all the wolves, the negative externalities of that action will become apparent, yet again, and the federal government will spend millions rehabbing the wolf populations until some populist conservative decides to decimate the population again. Rinse, repeat.
One problem with cattle is that they are being held in one place, can’t escape which makes them easy and reliable prey . So once a predator has figured that out they will come back more often. I have read about people having good success with dogs that are trained to confront predators like mountain lions and bears so they won’t come back.
Edit-but definitely start by paying them for whatever wolves eat.
Let's pump the brakes on torture box meat before we start trying to solve something many people don't even consider a problem (I believe ranching and free range meat are an almost unmitigated good. )
It's an appropriate use of tax money.
Reading https://www.minnpost.com/environment/2020/02/heres-how-much-..., that gets around 100 claims per year, for a payout of about $100,000.
The EU has a similar program (https://www.bbc.com/news/science-environment-46153727)
In Washington State, the newspaper articles on it always present it as one of:
1. the state sends out the helicopters and gunmen
2. the rancher eats the cost of the dead cows
I wouldn’t know whether that compensation is fair or how bureaucratic the procedure is.
Some studies have found that just "scaring" some prey animals reduces their breeding and encourages them to move around more.
However, I can't say we've noticed any reduction in the deer fecundity here because they get chased around. They simply learned to tolerate the occasional chase as a cost of living in our valley, which is evidently not too burdensome on them.
Any dog will chase them, and then the dog will also run out into the road. We don't want dogs doing that ;)
Putting all that into the Dog-O-Matic(TM) Computer....
... WORKING...
... beep boop Beep beep ker-chunk boop boop
... DING!
THE MOST SUITABLE DOG FOR YOUR REQUIREMENTS IS: the Scottish Deerhound[0].
====================
Of course, there are other dogs that would do just as well, like the Norwegian Elkhound[1], the Karelian Bear Dog[2,3], the Bluetick Coonhound[4, 5], or, even that well-known gentle giant, the Great Dane [6]. Believe it or not, Great Danes can run really, really fast. Their top speed is somewhere in the neighborhood of 30 mph[7]! Even Bambi might not be able to outrun that.
Whichever breed you choose, get your dog from a reputable breeder of working line dogs. Working dogs, show dogs, and pet dogs all tend to have different temperaments, and you'll have an easier time if you pick a dog from a line that's suited to his role. Frequently, the different lines will even have different physical characteristics[8, 9].
Not only that, but you'll want to make sure you pick a good breeder, and choose the dogs that will be most likely able to succeed at running down Bambi. For hunting dog breeds, at minimum, this means the dog's parents should have passed breed standard hunting trials, or the equivalent. The ultimate would be a pup whose parents are both field champions, or champions in some other doggie sport[11]. You want a star canine athlete, not a bench warmer.
Anyway...
TL;DR: Yes, breeding matters. No, you wouldn't have to create an entirely new dog breed to chase Bambi around. Pick a good breeder. Choose a dog based on his parents. You're looking for an athlete, not a pet.
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[0]: https://www.akc.org/dog-breeds/scottish-deerhound/
[1]: https://www.akc.org/dog-breeds/norwegian-elkhound/
[2]: That sounds made up, doesn't it? Seriously, it sounds like something straight out of Star Trek.
[3]: https://www.akc.org/dog-breeds/karelian-bear-dog/
[4]: Watch out when talking about your Coonhound in polite company. These dogs live to chase and corner raccoons ("coons") up in trees, barking at them from the base of the tree while the hunter shoots at them.
Why be careful? Unfortunately, "coon" is an extremely offensive racial slur against Black people in the US. You really want to avoid that type of association with dogs, Black people, and trees, if you know what I mean.
See https://www.ferris.edu/jimcrow/coon/
[5]: https://www.akc.org/dog-breeds/bluetick-coonhound/
[6]: https://www.akc.org/dog-breeds/great-dane/
[7]: https://highlandcanine.com/the-fastest-dog-breeds-in-the-wor...
[8]: https://www.allshepherd.com/working-lines-german-shepherd-vs...
[9]: https://www.allshepherd.com/german-shepherd-sloping-back-vs-...
[10]: https://www.akcchf.org/canine-health/canine-athlete-performa...
Just a thought