When I think of animal problems, I think of bear, mountain lion, dogs, moose in rut, wild pigs, alligators/crocodiles, snakes and poisonous insects. Most wild animals (except some dogs) generally prefer to avoid confrontations with people. We smell bad, are potentially dangerous, and are not their normal prey.
The general over all rules are:
1. situational awareness (joggers are often either concentrating on their running or in their "happy place"). Snakes, alligators, crocodiles, pigs, poisonous insects aren't likely to come hunting you, you walk up on them. LISTEN!!! I find ears work better than eyes in heavy brush (other way around, of
course, in open country). Don't talk or sing all the time, enjoy the silence, just make significant noise fairly often. If you realize you've moved in on something, move back, but don't turn your back or run. The primary weapon (defensive or offensive) we have is an engaged mind. If your mind is elsewhere, you are defenseless.
2. Let them know you're coming so they can avoid a conflict (don't ghost through the woods, make noise). Once you surprise them, some of these may run the wrong way, at you, so let them know your coming and they will generally just fade back out of your way and you won't even know they were there.
3. If a predator is nosing around, it's evaluating you. You want to discourage any animal as early as possible, while they are still in the 'is this dangerous or edible' stage of evaluating you rather than the "looks tasty" stage.
The goal is to avoid attack. If you are attacked, you want to deal with the beast before it reaches your tender body. Distance weapons, rather than a weapon for use once you're bloody. For that reason I think a knife is almost useless as a weapon. (It's still a great tool, always carry a knife!). If you've ever tried to stab or cut through a fresh bear or moose hide with the hair on, you know it would take some serious adrenaline to do much damage with a knife if your were attacked. (I know it's been done, I don't know of a case where the human came out of it without damage. I think the people in those cases were probably better men than me and terrified to
boot).
A gun is really the best protection in the event of actual attack. Not much outside of Africa can argue with a twelve guage slug within twenty feet.
For those who opt for less violence, bear spray (pepper spray) works and is cheaper and more compact (make sure it's quickly accessable), but it is only effective if a good dose gets in the animals nose and eyes. My brothers worked the Exxon Valdez oil spill clean up. They saw a bear run out of the woods at some workers. The bear guard sprayed the bear spray in the air and stepped back a few steps. When the bear hit the cloud of spray he went to the ground. He ran off and a few minutes later, returned with the same results. The third time he ran out, the bear guard lifted the can and the bear turned and ran back in the woods and didn't return. (note, bear spray is not a repellent. When they first came out with it some people sprayed it on their canoes, etc., thinking it would keep bears away. bears chewed up the canoe. Turns out, some bears like spicy food, just like some people, but no bear wants a nose full of it, just like people.)
An alternative to pepper spray I recently read about somewhere is wasp spray. It's pretty cheap, sprays a concentrated spray thirty feet, and can cause some pretty serious damage in the lungs in sinuses. A face full would be a serious deterrent for anything on two legs or four. (Yes, I know, it's poison, that's kind of the point). A large predator you hit will probably survive, but you would have taught it a potentially life saving lesson "don't mess with the smelly two legs, they hurt you from a distance". That is really the lesson we are trying to teach, if you are against hunting.
I remember once my son and I were walking down a trail near a salmon stream in Alaska during the salmon run, we didn't have a gun (yes, I know, stupid) when there was suddenly an overpowering stench of rotting fish. We froze in place. After about a minute, smell was gone. We never saw or heard the bear moving away from us. It was unnerving to realize how quiet they can move. We still had about 4 miles of trail to walk to get where we were going. We sang every song we knew (a lot of hymns for some reason) loudly the rest of the way.
dogs - This is by far the most likely problem in most areas I've been. Feral or just aggressive, free roaming dogs don't have much fear of people. They can get truly dangerous if they are in a pack. A single dog is less likely to cause problems. A staff is useful, a pocket full of rocks may be better, maybe with a slingshot. I want to discourage them before they get close to me. Most aggressive free roaming dogs I've known were very familiar with the motion of someone picking up a rock to throw. A related problem someone is bound to mention is wolves. They aren't that common and they mostly avoid people like the plague. If they start nosing around I would treat them like aggressive dogs. (Don't let dogs get behind you, that's where they prefer to attack from)
mountain lion - He's an ambush hunter, if he wants you, he'll hit you from behind with no warning. You will probably be on you face with his jaws around your neck about the time you realize he's there. If you're alone, you're probably dinner. Good news is that they are not that common and usually avoid people (If we still hunted them some, they would avoid us more. I think most of the places with lion problems are places where no one has hunted them for a long time.) Even if we didn't kill them, hunting them with dogs would teach them to stay the hell away from people. A dog, better 2 dogs, is probably your best warning system, but you want them trained to the point that you can get them to stay with you and not go haring off after something that will kill them, or worse, come running back to you bringing trouble with them.
bull moose in rut: Stay out of their way. They remind me of some guys I used to run with, so horny they can't think straight. They either want to fight it or have sex with it. They can be seriously bat-shit crazy! They even attack bears sometimes. They won't hunt you down, so just get out of their line of sight. Step behind a tree. They really aren't thinking straight and if you remove yourself, you'll be fine. A possible silver lining is that the rut often coincides with hunting season.
pigs, snakes, lion, bear - You aren't their normal prey. Mostly they just want to avoid a conflict. So be noisy, be loud, stomp around, sing songs. Don't ghost through the woods. Let them know you're coming so they can mosey out of the way. If you come up on something, try to fade back without looking like your are running away (predators and omnivores have a chase instinct). If you surprise them, brown/grizzly bears natural response is to attack (since they are top of the food chain). If a black bear is following you, circling you, etc. it is trying to decide if you are worth the risk of attacking (you are a pretty good sized animal you know, and vertical also, which makes you seem even bigger). Get as big and scary as you can, yell, bang sticks, throw things. Try to convince it that you WILL EAT IT if it gives you any shit. (Some people will recognize this as the proper response in some similar social situations). You may be saving the animal's life, because if it comes after you, you will need to kill it. (if you don't kill it, and it eats you, it's just learned that people aren't that tough and taste like pork, so someone will probably have to kill it even if you don't).
snakes, poisonous insects, alligators/crocodiles - stay the hell away from them. Don't be like Steve Irwin and 'poke em with a stick' (seriously, I loved the guy, but he is not the role model you want to avoid problems with animals).
lastly, as I mentioned, if it really comes down to an actual animal attack, a twelve guage or pepper spray are probably the best response.