From Nookipedia, the Animal Crossing wiki
The golden stag (オウゴンオニクワガタ, Ōgon'onikuwagata) beetle is tied with the Hercules beetle at 12,000 Bells as the most expensive insect in the game. Its first appearance is in Animal Crossing: City Folk. It reappears in Animal Crossing: New Leaf at an earlier time, starting at 5.p.m.. It is rare, and it can be found only on palm trees. It can be caught if the player approaches it as slowly as possible; being a rare, expensive bug, it is a skittish bug.
 Donating to the Museum
 In City Folk
Though less dramatically shaped than most stag beetles, this bug's gold, shiny color makes up for it! Another notable point about the species is the friendly way males and females interact. The most successful families are the ones that get along best, eh wot? Well done... vile bugs."
After donation, the golden stag beetle can be found on the second tier of the insect section in the first enclosure, walking on the tree stump towards the back. Once the Hercules Beetle has been donated, both will fight on the stump - lifting one another up in the air then tipping each other over.
 In New Leaf
Upon donation to the museum, the golden stag can be found on the stump in the upper right room of the Insect section. If the Horned Hercules is also in the museum, the two will be fighting on the stump.
"Golden stags are named after their unique gold color and are highly prized by collectors. They shine like gold when their bodies are dry, but when it's humid, they actually look black. Golden stags have short mandibles that aren't good for fighting, but males sometimes fight for territory. The males are surprisingly friendly to the females, which gives these beetles a likeable quality."
 Encyclopedia Information
 City Folk
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