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Doing a prequel before turnimg to Tigers. there have always been
Elephants.
And in a renewal year 2002 by Aronson figurative White Elephants:
http://www.chessvariants.org/unequal.dir/white-elephant.html.  A marauding
Elephant could be a predator notwithstanding the likely cause by that
creodont of the biological kingdom Humanity in either abuse or habitat
destruction.

Sittuyin of Burma has Elephant compounded of Ferz and forward Wazir, used
in the above CV White Elephant.  The Great Elephant is both that plus Alfil
plus forward only Dabbabah.

The Chaturanga Elephant, http://www.chessvariants.org/historic.dir/chaturanga.html, of historic proportion  moves as namesake Alfil thus restricted each to 1/4 the squares on majority of boards used including 8x8.

http://www.chessvariants.org/xiangqivariants.dir/tigers.html

Double-moving Chinese pawns.

This topic just started is for predator-named chess pieces. They have not
been organized together yet. Daniil and Charles show Tiger will take broad
coverage to include all the ideas.  So start with more straightforward Wolf
and Fox instead. (Of course to simple Orthodox players with simple RNBKQP,
Wolf and Fox are sophisticated.)  Neither Fox nor Wolf are leapers to any square making them very interesting. Unlike prosaic compound leapers, each is eminently blockable along pathway(s) by intervening pieces.

http://www.chessvariants.org/piececlopedia.dir/wolf.html.

Predator Wolf is doubly-bent rider.  To the Rook squares it is two-path;
there are two ways to get to Rook squares beyond the adjacent one.  Unlike
Sissa (http://www.chessvariants.org/piececlopedia.dir/sissa.html), which is also two-path to all Rook squares, Wolf does not reach
Wazir square.  Wolf and Sissa have different pathways to the orthogonal
Rook spaces, though both have two routes.  

Wolf's other squares seen in Stiles' diagram are single-path. Wolf reaches Alfil square interestingly, that is by omitting its optional middle leg.

Fox reverses the sequence of Wolf, each having three legs with middle optional: http://www.chessvariants.org/piececlopedia.dir/fox.html.  Fox then is two-path (as multi-path) to Bishop squares but cannot travel there as a conventional Bishop does.  Fox reaches Dabbabah square by omitting middle leg, and all its non-radial squares do include middle leg.

Notice the subtle difference that by logical complementary definitions, Fox can reach adjacent squares and Wolf cannot.  On piece values, since Fox reaches all the Bishop squares and has two ways to get to each one, Fox is certainly more valuable than Bishop, and factoring in the Dabbabah and oblique squares Fox also reaches, the piece value must be about 6.0.  Wolf is also somewhere near 5.5 - 6.5.

The intent is to be comprehensive of predator-named chess pieces. Still on
Wolves and Foxes, http://www.chessvariants.org/large.dir/evilhorde.html,
there in Fantasy Grand are two.  Werewolf moves forward as Knight, and
promotee Wolf Rider combines King and Knight.  Wolf has also been Carreran Champion Rook plus Knight and Fox Carreran Centaur: http://www.chessvariants.org/large.dir/wolf.html.  The so many languages, it's as if Arno von Wilpert were reaching out in 1940 to prevent WWII.

Charles Gilman implemented the Stiles' Fox and Wolf double-bend riders here: http://www.chessvariants.org/index/msdisplay.php?itemid=MS4linepiecefusi.  There he explains Rook plus Wolf is SheWolf, and Rook plus Fox Vixen.

No Giraffes of course.  But the ground is not shaky at all to call Gryphon predator and will include (all-spellings the one griffin, griffon etc.)

Do birds of prey count? Shogi variants are crowded with Eagles (Soaring,
Free, Cloud, Mountain-Hawk), Falcons (Horned, Flying, Raiding, Tenacious),
Owls, Dragons (Flying, Great, Reclining, Blue).

Also with Wolfs (Evil, Violent), Foxes (Enchanted, Treacherous), Badgers,
Bears (Blind, Violent), Lions, Tigers (Blind, Savage, White), Leopards

Thanks, all of them from the lead:
http://www.chessvariants.org/index/usereditcomment.php?commentid=31100.  
Hawk, Falcon, and the Shogi ones but those should be after the so many
Tiger and Lion first.  There are still more Gilman uses for Fox and Wolf
too (along with Shogi Wolves), and I think I can find another Werewolf before the Tiger and Lion
chapter.

Those named Wolf and those named Fox, as Lion, as Vulture, are not
arbitrary.  Something guides them.  Howl or utterance as vocalization well
convey information then meaning. It takes no semitic or mayan script to understand.   Early artifact Chess
nomenclature suggests lineage of consciousness, a reaction from the call of
the wild.  

Gilman's 'Man & Beasts' and 4 LinePiece Fusion expand on Stiles' Wolf and Fox as follows: 

Bishop+Wolf=WOLFHOUND, a creature that runs (think Laufer/Loper/Futar) after Wolves. 
Bishop+Fox=FOXHOUND, likewise after Foxes. This piece is colourbound. 
Wolf+Fox=WEREWOLF, a fairly obvious indicator of an enhanced Wolf. 

Betza's  http://www.chessvariants.org/other.dir/pasgl312-chess.html Fox is
 an up to three square Crooked Bishop plus unidirectional Wazir.

Shou Dou Qi Wolf is full Wazir, http://www.chessvariants.org/other.dir/animal.html, along with the others.

Christine Bagley-Jones description of Shogi units:
       Enchanted-Fox slides one or two squares diagonally
forward or straight backward;
Evil-Wolf description steps forwards, diagonally forwards, and
sideways;  name Running-Wolf slides any amount of squares diagonally
forward, sideways, or steps 1 square straight ahead, Taikyoku Shogi.

Regarding owls the next page of Man and Beast, page 14, has those, hough more from the bird aspect than the predatory one. The Barnowl has a SOLL of 39 and the Snowyowl and Tawnyowl 40 - in conjunction with the Flamingo and Stork having 37 and the Crane 40.