Virtually all dogs exhibit some degree of chase or prey drive. If, as with most dogs, the thrill is the chase - be it objects or other animals - and the dog shows no desire to cause injury or harm, then the behavior is pretty innocuous enough. In this instance, our biggest concern is being able to control the behavior, and thereby preventing accidents occurring - such as running across roads or worrying other animals. If this is typical of your dog, but the intensity of the chase instinct is such that traditional approaches to training have not been able to control him, then the remedy is similar to teaching the dog to come when called. But now we add another command, 'No'.
On the other hand, should your dog be of a predatory disposition, this is an entirely different and more pressing matter. When dogs engage in predatory behavior for real, it is aggression of a very serious nature, for the object is to kill.
From my long history of dealing with such dogs I can confidently state that one can never eradicate this desire to chase and hunt-down. This does not, however, mean that there is nothing we can do short of keeping the dog confined at all times. I have had great success using what's known in the trade as 'aversion therapy'. By a variety of means - one of which usually involves using an e-collar (electronic collar) - we can associate the act of predatory aggression with unpleasant consequences. For more information on the e-collar, see the link on coming back.
This is deadly serious business, whether the prey objects are cats, deer or livestock, and the behavior must, at minimum, be controlled.
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