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Possessive behavior in dogs is something that is easier to prevent at an early stage than to correct when it grows into aggression. When your dog becomes possessive, it's usually a result of insufficient leadership from your side. Your number one task should be to reclaim your alpha dog position.

Preventing possessive dog behavior

Preventing possessive dog behaviour is based on maintaining your pack leader role and reinforcing it on a daily basis. You are in charge, not your dog.
Set clear possession rules for the dog.
There is nothing in the house or in the street that belongs to the dog. That means, a dog should only touch things after your permission. If a dog wants something (food or a favourite toy), make it sit or lie down before it can have the desired thing.
Another good exercise is taking toys the dog is playing with away on a regular basis. This will teach your companion that you can do whatever you want with things even if they are in your dog's mouth.
Similarly, practise taking away the dog's food while it is eating. If you have difficulties doing this, step over the bowl and move into your dog's space slowly just like a dominant dog would do in the wild. Command your dog to sit or lie down, and give the food back only when your furry friend is calm and obedient.
Teach your dog the "leave it" and "drop it" commands.
These are very useful and very important commands. Besides the obvious purpose, they help retain your leading role. Follow the link above for training tips.

Correcting possessive dog behavior

Correcting possessive dog behavior is based on reclaiming your alpha dog position.
Never leave possessive or aggressive behavior without correction.
Small problems can result in big problems if left without correction. Correction is necessary even when a dog exhibits the slightest sign of possessive behavior. Don't wait until the behavior escalates to higher levels of intensity, which are always more difficult to fix. Some dogs may sound aggressive when they play but this is usually not a big deal as long as you can take the toys away and end the game at any time. If you cannot take a toy away, stop the game immediately and command your dog to sit or lie down, or any other command the dog knows well. This is necessary to redirect the dog's attention. If it still refuses to give you the toy, use a trick like offering your dog another toy. Do not snatch a toy away by saying "No!" because this doesn't teach the dog anything. Offering another toy will teach the dog that giving in one thing leads to another nice thing.
Don't avoid situations when your dog becomes possessive but create them and work on them.
If your dog is possessive about a certain toy, it won't help if you just hide the toy to never show it to the dog again. The dog will find another thing to become possessive about, including its food, and food is something you cannot just remove for good.
Instead, purposely create situations when your dog is likely to become possessive. Leash your dog to get more control and confidence, and drop its favourite toy on the floor. Correct the dog if it attempts to take the toy without your permission. If you do this many times a day, the dog will get the idea eventually. Repetition and patience are crucial. Always praise your dog profusely whenever it drops an object at your command and whenever it waits for your command before picking it. Positive reinforcement makes miracles.

If possessive behavior grows into aggression that makes you be afraid of your own dog and you are not sure how to address the problem safely, seek professional help.