Super-Persistent Predator



Our intrepid young adventurers are exploring their new unknown land for whatever reason it may be: be it money, pursuit of knowledge, or simply by accident (a quest for survival). In any case, they are unaccustomed to the land, but aren't exactly smart about being careful.

Namely, they run into some monstrous beast that wants only one thing from them: lunch. After a dangerous escape (someone will probably be killed), the explorers dust themselves off, maybe laugh nervously, and try to get as far away from that thing as possible. No harm done, time to focus on getting to the shelter or something, right?

But wait, what's that sound? Is it following them? Through rivers and mountains and who knows what else?

You bet. Seems like this thing just won't give up until it has another taste of human. No matter how long it goes without a meal.

Basically, this predator will hunt the protagonists far beyond the call of common sense or even instinct. Be it through fierce jungles, caves, canyons. You'll bet that, once the climax comes by, the beast will be right there ready for one Final Battle. If the heroes manage to find shelter, you can bet that the predator(s) will go through an extreme amount of effort to break windows, unlock doors, or learn to open complicated locks and then bash through a wall anyway to get to its prey. A situation like this might have you thinking that it'd be more productive to just find something else to eat. Sometimes it's explained away by stating that they've got human intelligence, but that just raises further questions about why an intelligent predator would expend so much effort to eat something that keeps eluding them. You might find an ecologist tearfully screaming that nobody, animal or human, would be this vindictively persistent just for one difficult meal.

But there's a reason this trope can work so very well. Note the word human intelligence. "Persistence hunting" isn't only a scarily good hunting strategy, provided the predator has enough stamina to keep chasing until the prey doesn't have the strength to keep running, it's also our strategy, both in hunting and escaping (waiting for the predator to give up). The use of this trope instantly makes the 'dumb animal' a credible threat and Worthy Opponent, since we don't have the option of outfoxing it until it gives up and goes away. No, this is something smart enough to see through our strategems, that won't fall prey to Attention Deficit Ooh Shiny and stop chasing us, and will continue the hunt even as we tire. It's clearly going to come down to a climactic battle of man vs. nature. Exciting, no?

Amusingly, this is one handwave that often wouldn't have been necessary -- it's a well known fact among biologists that intelligent predators can and will hunt for fun, even if they aren't hungry. For example, dolphins will torture and kill smaller breeds of dolphins as well as band together to gang rape young female dolphins - and males, they aren't picky; this goes double for orcas, which have been known to throw live seals around for hours on end, occasionally releasing them once they're done. Oh, and let's not forget about humans.

In practice, this is the animal version of the Implacable Man. The inversion is Animal Nemesis. Compare Attack! Attack! Attack!, when the animal fights irrationally. Usually happens in places where Everything Is Trying to Kill You. Biological equivalent of the Spiteful AI. The entire universe can be like this to The Chew Toy or Butt Monkey. If this beast appears at the end to chase the villain away, it's Exit, Pursued by a Bear. This trope has nothing to do with Predator, although that one is super persistent, albeit for different reasons.

Anime & Manga

 * Dragon Ball Z had an amazing example of this. Young Gohan was sent on survival training with little more than a sword. Eventually, he gets the hang of it. And apparently gets chased by a Tyrannosaurus Rex every morning. Not only does the T. Rex never get Gohan, every morning he knocks him out, and slices off part of his tail and eats it. And yet, despite his prey eating HIM every morning, he continues to hunt Gohan.
 * In the beginning, anyway. Eventually, it starts running away from him.
 * Revolutionary Girl Utena featured a herd of super-persistent elephants. A Crowning Moment of Funny for the part where they're chasing Nanami's raft across the ocean on surfboards.
 * Justified/Invoked with the Abyss Eaters, from Claymore, as they are conditioned to crave the flesh of one particular target. Said target is much, much stronger than them, but their persistence wears him down untill he dies a Death of a Thousand Cuts after months in a constant state of battle.

Film

 * This trope is the premise of The Ghost and the Darkness.
 * The lion in this movie never actually chased or hunted them to a ridiculous extent. The humans had just moved into its territory (building a railroad I think) and it was picking them off for food. This happens in Africa in Real Life. A hungry lion will repeatedly attack a small village because it learns that humans with only mud huts for shelter and primitive weapons are easy prey. Usually the lion won't stop picking off people until it is killed.
 * In fact, it is specifically stated that the lions would kill even when not hungry.
 * This is also a true story; the lions are stuffed and on display in the natural science museum in Chicago. Recent studies have suggested that between the two lions, they only ate some 40 people, not the 140 originally claimed.
 * Oh boy, pick a Sci Fi Channel "Original Movie". To name a few of the more unintentionally hilarious examples:
 * In Kaw, Ravens and Crows get infected with Mad Cow (?!) disease after eating clearly diseased rotting cow flesh. As soon as they go crazy, they decide to exclusively go for the human protagonists (not even any other areas, just those few people). This includes waiting on a bus while the the humans cry and then throwing rocks at the bus in a desperate attempt to get inside. Later, they slam themselves against a diner in order to get in and eat the people inside, before they inexplicably die. You'd think there weren't any animals in the forest. Subverted in that the super persistent predators starve to death, just as they would in real life.
 * Shockingly averted in the (surprisingly good) reimagining of The Land That Time Forgot (which starred and was directed by C. Thomas Howell) in which the Rex only chases the humans when they enter it's territory to steal food, and all it does is just try to run them off.
 * Used as a plot point in Sharktopus - one character wonders why the eponymous creature is going out of its way (such as attacking boats and going over land) to attack humans rather than just hunt aquatic creatures, and it turned out there had been secret changes to its genome to make it a better killer.
 * Justified in Crocodile (but not so much the Sci-Fi Original sequels third and onward), featuring a crocodile (giant, naturally) pursuing a group of half-dressed teenagers. One of them had one of her eggs in his backpack, fueling her maternal rage. The survivors are allowed to leave when they return it.
 * Not really. Though crocodiles actually do have fairly strong protective instincts, they wouldn't chase after someone. If a nest robber got away from it, it's not going to bother to chase it down.
 * It's implied to be a sapient, magical demonic crocodile.
 * Considering it was not a reproductive queen, the xenomorph in the original Alien film sure went to some lengths to hunt down and capture (or kill) the crew. Probably justified in the fact that the queen/worker/hive analogy wasn't developed until the second film.
 * Justified in deleted scenes. It was using an alternate method to make eggs. That being 'the people'.
 * One theory raised by those making the movie was that the aliens were created as weapons of war, so that would also make their Kill'Em All actions plausible. They're also strongly implied to be considerably more intelligent than any ordinary predator, giving it a possible motive beyond food.
 * It's not like there would have been all that much food in the Nostromo's air ducts; it might just have been stocking up food. And it might have percieved the ship as its territory or the humans as potential threats. Or some combination of the above.
 * It felt more like it was getting them stocked up until the ship hit land again....great, now I'm wondering if it could pilot a ship....
 * The truck in Duel behaves like this just For the Evulz.
 * The shark in Jaws, more so in the sequels.
 * To ridiculous extremes in Jaws: The Revenge. The shark apparently tracks down people over hundreds of miles in order to kill them in revenge for their father having killed a couple of other sharks a decade earlier.
 * Jurassic Park teetered over the edge of this trope for two films:
 * In the first, the T. Rex and the "velociraptors" aren't following the humans as much as chancing upon them and chasing them a bit. The first encounter with the T. Rex plays this trope quite straight, giving up only after any real animal would've given up, and it's doubtful that dinosaurs chased low-fat prey to extents where the gained energy would not nearly cover that expended, not mentioning that at the speeds that the jeep is going a Tyrannosaurus would be suicidal to run at, let alone have the physique for. However, both the T. Rex and the raptors in the original novel are extremely persistent.
 * Justified in the second: the Tyrannosaurus couple is chasing the humans because they had kidnapped their baby, and one of them is still walking around in clothing smeared with their infant's blood.
 * In the third, however, they just throw reason to the wind, and have all the dinosaurid predators behave this way. The Spinosaurus chase the protagonists about a mile farther than reason would allow. One would think that there weren't any other dinosaurs around, or that it was just that cruel. They also stumble upon a Tyrannosaurus standing over a dead dino (looks like it's scavenging) with plenty of meat, but when it sees the humans, it naturally starts pursuing them instead. Which incidentally leads it to the Spinosaurus, whom it engages and dies. It might have perceived them as a threat to its meal and be chasing them off... in which case it ought to have stopped chasing them as soon as they were out of sight and gone back to its food. The Velociraptors of this film, however, were chasing the heroes because.
 * While all of the predators of King Kong's Skull Island seemed to want to try out the new taste sensation of humanity more than the next one, none were worse than the T-Rex T. Rex V. Rex who actually runs after Ann Darrow with the corpse of a current kill in his mouth. Some of the V. Rexes on the island actually sacrifice themselves in their attempts to kill her. And when given the chance to bite Kong versus swallowing Ann, they always went for Ann.
 * In the first V. rex's defense, the creature it killed is described in Natural History of Skull Island as a disgusting, smelly crocodilian scavenger with foul, pestilent drool like a komodo dragon's. It's possible that the carcass the rex had been chewing on tasted horrible.
 * Not to mention the Venatosaurus raptor-pack, which idiotically tried to chomp humans in the middle of a Brontosaurus stampede, then persisted in chasing down hapless cameramen, rather than gorging on several thousand tons of fresh bronto meat that was lying there waiting for them.
 * except its actually justified whats easier to kill, a kiloton animal that can crush you with its food, a human which has no defense(other than a machine-gun but still),
 * Keep in mind that there are other humans on Skull Island, presumably hunter-gatherer types who may hunt some of the same animals the dinosaurs eat. They probably have an instinctive hostility towards humans because top predators often kill each other to reduce competition for food.
 * Going by the The World Of Kong: A Natural History of Skull Island tie-in book, the human Skull Islanders are well aware they're stuck at the bottom of the local food chain. They hide out on the coastal side of that giant stone wall because they know from experience they'd be snack food in the dino-infested interior.
 * The sharks in Deep Blue Sea. Tom Jane even points out that sharks don't particularly like the taste of people. Justified because
 * TV B-Movie The Last Dinosaur has the titular titan, a T. Rex with Implacable Man tendencies, treats a group of explorers like this...at first. Then we see it kill and eat pretty much everything unlucky enough to cross its path. Implicitly it's eaten most every other animal in the Lost World.
 * Used as padding in the new Star Trek film. Kirk is being chased by some shambling furry thing. Then a giant red ant bursts out of the ice, bites the furry thing, tosses it aside...and starts chasing Kirk. Even though the shambling furry thing was bigger (but still bite-sized to it) and it had already attacked it.
 * Makes significantly more sense if you interpret the red creature as being territorial as opposed to hungry. It frequently stops to roar at Kirk, something a hunting animal is unlikely to do but an animal trying to intimidate a supposed threat might, and
 * Which makes significantly less sense once you take into account how long it chased Kirk, and how quickly Kirk established he wasn't a threat.
 * Bigger animal, bigger territory. Also, chase instinct.
 * Maybe it was a little of both, it killed the furry beast and saved it for later
 * And another small note, it didn't intend to chase Kirk down the ice cliff, it only came down because the cliff broke and it slipped.
 * Not exactly a predator, but the prehistoric squirrel in Ice Age is pretty persistent about getting That One Nut; his attempts, and subsequent failures, to eat is a Running Gag.
 * And in the third film, we have.
 * The bear in The Edge was a perfect example of this trope.
 * The killer whale in Orca the Killer Whale. Justified in that it was out for revenge, something dolphins regularly do in Real Life, though in the film it was portrayed as a male avenging his mate when in reality it would be a whole pod avenging one of their friends.
 * Wife and child. Which is why Orca is more like an aquatic Death Wish than a Jaws imitator.
 * The crocodile from Peter Pan. Justified with the bit of backstory that the croc had eaten Hook's amputated hand and liked the taste so much that he wanted the rest of the dish.
 * The shark which manages to flop very fast across an unclear distance of land and right into a volcano in pursuit of the very small lemur Mort in Madagascar 2.
 * Played with in the 2009 Land of the Lost movie. The resident T-Rex, Grumpy, actually is all set to give up on chasing Marshall, Will, and Holly after they prove to be more trouble than they're worth. That's until Dr. Marshall insults the T-Rex's intelligence, causing it to hold a grudge against him for the rest of the film.
 * Anaconda combines this with You Fail Biology Forever, stating that the anaconda is some kind of Blood Knight that enjoys killing so much that it will regurgitate its latest meal just so it can hunt and kill again. Needless to say, no real reptile would waste energy like that. Anacondas and other giant snakes were once thought to do that, it turns out they only regurgitate their food if they're overly threatened while lethargic (btw, the snake in the movie should find a safe place and go into a self-induced coma after eating just one guy, nevermind a whole boat), or because they're, y'know, getting sick off it.
 * The sequel gets around this by there being many snakes in the same small area due to unique circumstances. It's even pointed out that a single one shouldn't be a problem after it has fed itself for quite a while.
 * The titular pigeon from Pigeon: Impossible as shown here - Allowing a nuclear missile to blow up another country, all for just a bagel.
 * Perhaps by far the most ridiculous example, the baby Godzillas in the American film. The fact that they so persistently attack the humans has several problems:
 * As newly hatched animals, their first priority should be to feed themselves. Since there is an abundance of available fish, the vast majority would be focused entirely on them.
 * As newborns, they should be starving and exhausted from the effort of hatching. This would mean that they would not have the energy to chase around and overpower humans (to the point of tearing their way through metal doors to get at them), especially considering the abundance of prone and helpless fish available.
 * There are literally hundreds of them, so the idea that they should all start competing over an incredibly small number of people to eat (even at some points seeming to start fighting among themselves over who gets to eat the people) when they can easily just break off the chase and go for the fish (even if a small number of them continued chasing the people) makes no sense.
 * A Hand Wave is apparently attempted when Nick states that they smell like the fish. Even if they smelled strongly enough like fish to be distinguishable among the hundreds of present and available fish that the baby 'Zillas would want to eat them, if the only thing which qualifies them to be attacked is smelling like fish, than they should have a greater preference for the more abundant and easier to eat creatures that smell like fish (namely, THE FREAKING FISH!!!)
 * Its mentioned a little while into the chase that the babies have already eaten all the fish and they will grow quickly. This troper assumed that the hundreds of babies quickly ate all the fish and were still hungry, so chased and fought amoung themselves for the next available food source: the people.
 * That's a lot of fish.
 * Last stretch of Prophecy has the main cast dealing with a literal (mutated) Mama Bear that wants to get them no matter what.
 * The Tiger in Burning Bright
 * Justified in that the Tiger was starved for nearly a week, and there was literally no other food supply.
 * The Graboids in Tremors are willing to wait out for days, weeks, months to catch a prey marooned in high ground. It doesn't help that they're quite clever and intelligent, and can find means to get to their prey, even though they have no eyes.
 * The movies all take place in desert environments, which tend to be home to small lizards, snakes, and rodents, and relatively lacking in other large prey. It's not inconcievable that a graboid might find it worthwhile to camp out in one spot or even chase humans, which would be a significant sized meal.
 * Justified in the TV series, where Jodie remarks that Graboids, or at least El Blanco, literally do prefer eating people over other potential prey.
 * An inversion in Cool Hand Luke: when Luke escapes prison, he runs so persistently that the bloodhound trailing him runs itself to death.
 * The Albertosaurus from the docu-movie March of the Dinosaurs. It pursues the small Scar and his sick companion even after And to think it just could have gone back to its pack who had literally tons of fresh meat at the ready.
 * The sea monsters in Deep Rising, who continue to relentlessly pursue the heroes despite suffering extreme gunfire trauma from doing so every time. One might also wonder why.
 * Played for Laughs in The Gods Must Be Crazy II, in which the predator is the small honey badger and the prey is a guy (and later, his boot).
 * The Grey features a pack of vicious wolves as a central plot point.
 * Justified, because.
 * Justified in Red Planet with AMEE, a dog-like robot borrowed from the Marines for the mission, whose processor gets damaged in the crash, and it classifies the astronauts as "foe". Not being an animal (or a living being), AMEE doesn't need to rest, feed, drink, or sleep. It runs on a fusion battery, so power is not an issue. It can chase them as long as it has to. AMEE follows classic guerilla tactics. The first thing it does is wound one of them to slow down the whole group and then follow, using its aerial drone to spy on the humans. Whenever a member of the group is separated, it moves it for the kill, making sure to broadcast the video of the murder to the rest of the group to demoralize them and incite panic.
 * Maximus the Horse from Tangled goes to incredible lengths to chase Flynn Rider, even getting into a swordfight with him.

Literature

 * In Michael Crichton's Jurassic Park, Muldoon commented that the raptors were cruelly intelligent and liked to hunt for sport as much as for food. It was actually justified in the second book, which explained that because raptors were so intelligent, being born and raised without actual raptor "parents" to raise them properly turned them into violent and chaotic creatures.
 * Actually, the raptors weren't a big problem until the end of the book, implying that they didn't realize they had access to the humans-only areas for quite a while. Also, since they tried to attack humans earlier, they probably already recognized humans as a credible threat to their survival and chose to eliminate them all the first chance they got.
 * Happens in an utterly over-the-top way in Wraith Squadron, with an insect that supposedly follows mammalian prey it encounters... well, pretty much as far as it has to. (Even managing to sneak onto the heroes' spaceship.) The Storini Crystal Deceiver is said to paralyze its prey and eat them alive; victims can be saved if they are found before too much biomass has been devoured.
 * Justified in the Artemis Fowl series with Trolls. Apparantly to them, humans are by far the most deliciousest things ever, and after trying one, a troll will do anything to eat more.
 * Also doesn't hurt that they're dumb as rocks.
 * There's also the fact that, as subterranean predators, they have extremely finely-tuned sense, so that exposure to the surface tends to cause them sensory overload which causes severe brain damage, usually removing most of the survival instinct.
 * In one of the Animorphs Megamorphs books (a special event book in which all six characters narrate) they are transported back in time and find themselves battling (of course) a Tyrannosaurus Rex. While fleeing they morph to escape, and Marco morphs into an osprey, his standard bird choice, that has both the ability to fly and should logically be too small to keep the predator's attention. However, instead of giving up once the already-small meal becomes an even smaller meal, as Marco logically expects it to do, the dinosaur keeps chasing him, and actually begin to tear apart trees that are standing in the way. Marco ultimately realizes that food has stopped being the point of the chase, and that the Rex is chasing him out of pure blood-lust and rage.
 * While they are humanoid, the trolls in R.A. Salvatore's Streams of Silver act like predators (they are trying to kill and eat the heroes, after all), and they are extremely persistent.
 * In Piers Ant hony's Orn, one of the characters has a prolonged battle with a Tyrannosaurus, in the sense that they play an extended game of cat and mouse. The in-world justification for this is that the dinosaur has evolved the tactic of pursuing its meals relentlessly, thus ensuring that, eventually, it will eat.
 * Which is still stupid, because the whole reason predators will eventually give up is because there becomes a certain point where they risk burning more calories trying to catch the prey than they'd get from eating it so it becomes better to just conserve their energy.
 * Captain Ahab.
 * Nately's whore.
 * In Robert E. Howard's "Red Nails" Conan the Barbarian knows that the dragon having treed them will wait forever if needed.
 * The Wisteron in The Elf Queen of Shannara which puruses Wren and the other Elves any time they so much as put a toe outside of Arborlon, and flatly refuses to stay down. Justified, as like the other Shadowen on the island, it hates the Elves for creating it in the first place.
 * The Graak and the caulls in The Voyage of the Jerle Shannara also behave this way. It's justified in the case of the caulls, which are magically-altered Mutant wolves designed to track their victims relentlessly. The Graak, a massive, bloody-minded dinosaur doesn't have any justification for its behaviour but is such a Primal Fear that it doesn't really need one.
 * The Tracker Jacker wasps of The Hunger Games. Their name is pretty self-explanitory, as they were bred to A) track down their prey and B) (hi)jack their victim's nervous system with their Grade A hallucinogenic venom.
 * Gone to absolutely ridiculous and eventually cinematic extremes in David Fletcher's Hunted: A True Story of Survival, in which a literal Mama Bear goes after the writer for killing her cub and it only ends when the writer manages to crush the bear under tons of ice. The kicker? It's supposed to be based on a "true story"

Live Action TV

 * In Primeval if it is carnivorous and came through an anomaly, then it is going to be an example of this trope. If it's not carnivorous, it will still attack you, just not eat you.
 * Not especially. The Smilodon was the only carnivore that tried to chase prey for long distances (very much at odds with what we know of the real animal, which had relatively short legs and heavy muscle attachment sites, indicating that it was a sprinter, not a distance runner). Other than that, the carnivores tended to attack whatever was near them, not pursue specific targets for long periods of time (the Gorgonopsid arguably being an exception, depending on whether it was supposed to be deliberately chasing the kid or just randomly found him again).
 * Several of the cryptids featured in Lost Tapes play to this trope.
 * Inadvertently deconstructed in the Babylon 5 episode "Grey 17 Is Missing" with a craptastic alien "predator" called a Zarg straight out of Mystery Science Theater 3000. It wasn't stealthy, it wasn't intimidating, it wasn't even particularly original, and producer Straczynski has been known to say he wants to apologize to fans in person for it.
 * Played scarily straight with the Shadow Battlecrabs. Once they lock on a target they will never cease pursuit until it is destroyed. Seeing how they can easily overtake almost any other spaceship in the galaxy, they rarely have to resort to this.
 * Terra Nova plays with this. In one sequence, a pair of Carnotaurus chase two jeeps up to the gates of the titular settlement, but when TN's defensive sonic weapons start firing, the dinosaurs decide to look for something easier to kill. The trope is later played straight with the fictional Slashers, which are really, really determined to get at the stranded kids in the broken down jeep, never mind those pesky assault rifles the kids have.

Radio

 * In the Doctor Who radio serial "The Paradise of Death", the Gargan is a predator that ignores you unless you enter its territory (which it thoughtfully marks with rocks). If you do pass the rocks, it pursues you until it has eaten you.

Tabletop Games

 * Raksha (souleating Lovecraftian fae from Exalted) find human dreams and emotions especially tasty.
 * Likewise, there's Zsofika the Kite Flute, a Second Circle Demon known for choosing one target and chasing it every chance she gets until it's dead. It doesn't matter if you banish her or dispatch her back to Malfeas via copious stabbings - unless you find some way to permanently kill her, as long as she's in Creation and she's not bound to a task, she will be coming after you.
 * In Paranoia, a doberbot's combat programming works this way. "Attack someone until he dies. Attack someone else until he dies. Attack someone else until he dies..."

Video Games

 * Alma from the Adventures of Lolo games. She'll normally mind her own business, but if Lolo should get within her field of vision, she'll roll after him until either she catches him or if he takes refuge in a meadow.
 * Many of the monsters in Final Fantasy XII seem to go to rather crazy lengths to catch the party. The ones that teleport however, won't leave you alone until you leave the room, and a few monsters might just keep going.
 * There was a glitch in The Oregon Trail III that sometimes made wild animals act like this. If you went out hunting and fled from an angry bear, the bear would appear right in your face the next time you went hunting, ready to maul your party members. This bear would follow your wagon for thousands of miles until you managed to kill it.
 * While Peter Jackson's version of King Kong uses this trope full-force in the movie as seen above, and you still get chased way too far by the V. rexes, the Official Game Of The Movie actually averts this as a specific game mechanic. A lot of time, you can divert the attention of a predator away from you by killing something smaller, causing the larger enemy to take the easier meal. There are even giant dragonflies and grubs you can stab with a spear, specifically for creating such distractions.
 * Most enemies in video games, really. Only rarely the AI even includes an option for the enemy to give up the chase.
 * The walrus chef in Crash Bandicoot: Twinsanity, who chases you halfway across N Gin's battleship - over seemingly arbitrary holes in the floor, through walls of crates - even nitro crates don't slow it down.
 * Nemesis from Resident Evil 3 Nemesis deserves special mention as he can follow you even to other rooms. Normally all one had to do to run away from an enemy is go into another room, but you have to run away quite a bit from Nemesis before he finally gives up.
 * Except the Nemesis has a perfectly justified reason for this, as it was explicitly engineered and programmed to hunt down STARS members, which Jill is. That's literally all it was made to do.
 * Monster Hunter inverts this. Most large monsters tend to flee from the player after exhausting themselves or sustaining large amounts of damage, and usually have to be chased in order to be taken down. In other words, the player is the Super-Persistent Predator.
 * Limbo has the game's only boss, a giant spider.
 * Justified in Vampire: The Masquerade - Bloodlines with the werewolves. specifically decides to hide out in their territory because they're known to hate vampires, and no one will look for him there. Even then, they leave him alone, until --then two of them go looking for retribution.
 * Metroid Fusion has the SA-X, Samus' Power Suit taken over by an X-parasite which chases you through the entire game with no way to stop it.
 * Averted in Fallout 3 of all things. Some predators will only chase you a relatively short distance, others have path finding issues and get lost.
 * Let's Go Jungle: Lost On The Island Of Spice has that oversized tarantula. It chases you as the first boss (but your characters escape it), then re-appears in the third level as a Mini Boss (and your characters knock it off a cliff), and just at the end, it attacks you again. You finally put it to an end by dropping it on deadly radioactive waste, which causes the plants nearby to mutate into a Man-Eating Plant.
 * Pokémon has Krookodile, a crook crocodilian that never allows its prey to escape.
 * While never specified to be a predator, Primeape fits the persistent part because it never gives up chasing whoever angers it till it has beaten them up. This happens a lot, because the Pokemon has such a extreme Hair-Trigger Temper it's near constantly in a Unstoppable Rage.
 * Averted in Jurassic Park: Operation Genesis which, as a Simulation Game, tracks hunger and other needs of all the dinosaurs. As such, the carnivores don't hunt prey any more than they (semi) realistically should. One reviewer was very disappointed when he released a T-rex near a crowd of people and instead of going on a massive rampage, the T-rex just gobbled up one nearby human, and then wandered back into its habitat and took a nap.
 * The Gyorgs around Tingle Island in The Legend of Zelda the Wind Waker will follow you until you either reach land or kill them.
 * In Oblivion, basically every enemy is this. That mountain lion, that wolf, that troll, that rat will chase you down to the ends of the earth in order to maul/feast upon your flesh. They will come after you through villages, forests, rivers, lakes, mountains and plains, from the farthest western point of the map to to the farthest eastern point, in an all-consuming, single-minded drive to wreak vengeance on you for entering their line of sight.

Web Comic

 * The Pteranodon birdosaurus in The Adventures of Dr. McNinja.
 * In Issue 06 of The Beast Legion: Xeus's Were Tiger form keeps coming at Master Surya despite all the obstructions.

Web Original

 * A rather terrifying example, the SCP Foundation features SCP-096, a normally harmless creature until you see its face. It then flies into a rage and will stop at nothing to kill you. Military-grade armaments have little to no effect during its pursuit of a target. The effect also works on pictures, and its face could be as small as four pixels and it would still make all attempts to brutally [DATA EXPUNGED].
 * College Humor and Harry Partridge's BearShark. Without a doubt.
 * DR. OCTAGONAPUS BLAAAAAAAAAARRGGHHHHH!!

Western Animation

 * Sharptooth (Tyrannosaur) from the first The Land Before Time hunted the dinosaur heroes through mountains and a desert, though this may just have been because there was nothing else to eat.
 * Alternatively, it could have been a particularly cruel and sadistic Sharptooth, given that all the other dinosaurs are at least semi-sentient.
 * Revenge may have played a part, seeing as the Sharptooth got his eyesight damaged and fell to his near-death during his first scrape with the youngsters.
 * It's actually revealed in one of the sequels that Sharpteeth are intelligent enough to talk (in a language the herbivores can't understand); they're just jerks.
 * What sequels?
 * Sylvester. Who, in one of the clip shows, has chased Tweety across the world.
 * Not just a clip show. There is a movie about Tweety going around the world in 80 days, and Sylvester chased Tweety through the entire world.
 * Tweety deserves it.
 * Poor old Wile E Coyote, chasing the Roadrunner far beyond the call for reason, going so far as to buy countless Acme products and mountains of birdseed instead of spending that money on something he could actually eat. It actually was explained by Wile E. himself, in the only episode in which he speaks. He addresses the question of, "Why would a supposedly intelligent predator invest so much time and energy chasing a difficult prey with very little meat?" He shows a diagram of the roadrunner and how its different cuts correspond, in the coyote's palate, to the most sumptuous delicacies that humans enjoy.
 * It must also be added that, apparently, there's no other source of food in the middle of the desert where Coyote and Roadrunner live. Oh if only there was a company that offered a service of moving products from one part of the world to another....
 * One of the Rules for the Coyote and Roadrunner cartoons set in place by Chuck Jones, the rules that made the cartoons so awesome, is that "The Coyote can give up at any time." Being a Super Persistent Predator is a choice of his own, maybe even an addiction he just can't quit, no matter what he does. Which is morbidly hilarious.
 * The hunt for the Roadrunner as life purpose for Coyote is played in this Seth Mc Farlanes Cavalcade Of Cartoon Comedy episode.
 * Tom of Tom and Jerry, whose obsession is best relayed through the fact that he once chased Jerry into a dog pound.
 * Like Tweety, Jerry deserves it.
 * In the Robot Chicken sketch involving the Smurfs getting killed in a flood has Gargamel, after years of trying to capture the Smurfs so he can eat them, finally being able to eat them (what with so many Smurf corpses). When he takes a bite though, the look on his face is that of "I've wasted my life".
 * In the original comics, he doesn't want to eat them (Azrael does), but use them as ingredients to create a philosopher's stone. But like other cartoon villains mentioned above, he is growing increasingly frustrated and angered with the Smurfs foiling his plans so sometimes he just wants to make them suffer.
 * Yet another Robot Chicken sketch (the Se7en parody) also lampshaded Gargamel's wobbling between the two motives.
 * The Smurfs are a versatile resource for Gargamel.
 * The Garfield Show - A wolf chases after John, Odie, and Garfield riding in a car for twenty miles until it attacks them in their home. Justified because the wolf was a mother who wanted its baby, which Odie adopted and took with him after the picnic trip.
 * And the panther in the special Garfield in the Rough, which attacks them at their campsite and smashes the driver-side window of Jon's car with its head (taking two tries to break it), then turns to go after Garfield after he drops on its back and claws and bites the panther.
 * This trope is somewhat true of Sabor in The Legend of Tarzan ; she only appears a couple of times, but both times she is incredibly single-minded about her chosen prey. If I were a leopard waiting to attack a baby and a gorilla showed up to protect it, I'd go find something easier; not Sabor.
 * To its most ludicrous extreme: In Kingdom Hearts she repeatedly stalks and attacks a boy who can shoot fire and lighting from his body.
 * Carnotaurus, the main villain of Dinosaur.
 * The leopard seal from Don Bluth's The Pebble and the Penguin.
 * A panther in one episode Timon and Pumbaa chased them into a tree, where the duo decided to wait for the predator to become disinterested and leave. They're frail, old men with canes and walkers by the time they decide to leave, and the equally-old and frail panther still wants to chase them!
 * Zig the hyena from Zig and Sharko spends every episode going through outrageous lengths and a ton of Amusing Injuries in hopes of capturing and eating a Mermaid.

Real Life
"Regular bees will give you about nine seconds of being too close to the hive before deciding you're a threat and then attacking you. So it's pretty easy to just walk past them without any screams. And if you do get them after you, they'll consider you to be 'chased off' after about 300 feet.
 * Many predators really do behave this way, though not to the ridiculous extents often depicted in fiction. The constant pursuit denies the prey a chance to feed, drink, or rest. Eventually the prey will either collapse or be too weak to defend itself. This can easily be sped up by injuring the prey or forcing them into dangerous terrain. These tactics can allow a slower, weaker predator to take down big game. The predator runs the same risks, so this can often be a make-it or break-it tactic.
 * This tactic is usually employed by a group of smaller predators pursuing one large prey animal, whereas many movies and TV shows have it reversed (one super-predator tracking a group of puny humans).
 * Humans are perhaps the most successful example of this trope. We are nearly tireless by the standards of most other animals, though hardly fast animals -- in our hunter-gatherer days, our favored tactic seems to have been following an animal at a jog until it simply dropped of exhaustion and heatstroke and either died there or had its head bashed in with a rock.
 * The Other Wiki has it listed under Persistence Hunting. This is still practiced by many African tribes (Before you try it on your local whitetails, though, remember that these people are hunting in vast expanses of flat land; there's no forests for the animal to hide in or hills for it to vanish over.).
 * People often talk about humans as being slow and weak compared to other animals, however in reality our bipedal locomotion and long legs allow us to travel longer distances without rest than most other land animals, so providing you are able to track your prey well (which humans can thanks to deductive reasoning and theory of mind) this is a pretty damn effective way of hunting.
 * Not to mention that humans know how to hunt in packs, and can thus take turns as the point man as well as using flankers to herd the prey in the desired directions. They can also carry rations, thus hunting for days at a time, and if the prey does escape it doesn't mean immediate starvation. Also human hunters can hunt not just for food but because they are just feeling ornery.
 * The Tarahumara tribes of Northern Mexico have the Kalahari (the main african tribe that still hunts like this) beat. A common game for them is essentially a group race where teammates pass a ball to each other while running towards a general goal. The short races are a couple hours nonstop. The long races? a few days (and its still nonstop, they walk for a break)
 * To use an example that isn't subsistence hunting, consider what happens if a given individual animal is considered a threat to humans: people will be quite willing to do what it takes to track down their target and remove the threat.
 * As noted above, this behavior is thought to be one of the reasons why other "Predators" (everything from Sabertoothed Cats of Old to a neighbor's dog that bites) disappear soon after humans appear in an area. Lions, arguably the world's most famous bad ass predatory land animal in the world are an endangered species and humans are not for a reason. Apparently no animal on Earth is as persistent or bad ass as humans on the hunt. The sheer amount of time and effort humans are willing to invest on killing is truly unparalleled in the animal world.
 * Besides humans, there is another well-known pack animal which uses persistence as its primary hunting strategy: wolves. Kind of makes sense that these two species would have gotten along fairly well in ancient times, doesn't it? Mans Best Friend might have originated as a pack of wolves and a pack of humans getting to know each other while chasing down the same big game, teaming up to take it down, and sharing in the kill.
 * Speaking of canines, African wild dogs are persistent hunters. They will chase their prey until it tires out, sometimes for miles.
 * Hyenas are also known to do this.
 * Cracked.com's "The 5 Most Horrifying Bugs in the World" describes killer bees as follows:

Africanized bees do not roll this way. They give you half a second of being too close before they decide it is time to completely fuck your shit up and empty the entire hive--tens, maybe hundreds of thousands of angry, angry bees. When you run, flailing and crying and soiling yourself while screaming "JESUS CHRIST I'M COVERED IN BEES," they will chase you for over half a mile."


 * Justified, because chemicals emitted by the detached stingers of the bees who already stung you attract the rest of the hive to continue the assault.
 * "The predator that won't stop chasing you" is an extremely common dream scenario.