You should know how this works - now that the obligatory expedition is complete, the area will more or less function like any other region. The monsters inhabiting it will vary, it's various resources may have Upsurge or Flourishing events, and the researcher that haunts the place will rotate out. This section, then, is to cover the new tasks these three researchers give in the Elder's Recess. In addition to that, there are unique items to gather from Beryl Deposits and Amber Deposits that only appear when the Elder's Recess is affected by the appropriate "Upsurge" event, which will unlock more ingredients at the Canteen. No reason not to buff up your meals, right?
Cultural Exchange: Gajalaka Linguistics II
Make an expedition to the Elder's Recess and wait until night time - there should be a blue moon icon in the top-left of your screen - then head down to AREA 8 to find the Lynian Researcher. If it's not night, he'll tell you to wait. If it is night, however, he'll warn you about Gajalaka guards ahead. Your goal is to infiltrate and meet their leader so you can parley, but overly-eager guards may prevent that. Looks like you'll have to sneak your way past the guards, and as the Lynian Researcher and your Palico suggest, you can wait until they're "dancing" (read: looking away), or you can grab your Ghillie Mantle, which will make the process much, much easier.
Sneak past the first guard, then use some crystals as a bridge, crawl under another crystal, then sneak past a second guard to reach the Gajalaka camp, then sneak into a small cave to reach the Gajalaka leader. After some chatter between the Lynian Researcher and the Gajalaka, you'll secure yourself an alliance. The Gajalaka will now join you if you meet with non-hostile ones on this map, and you'll have access to their various tunnels, allowing you to fast travel across the map (just open your map to find them - they're number-coded, not color-coded, for some reason). If that's not enough, you'll find yourself in AREA 16, a new location, and as a symbol of your friendship the Gajalaka will give you the Meowlotov Cocktail Palico accessory.
If you still want more reward for this task, you're in luck! Talk to your Palico at camp and it'll clarify about which Gajalaka will join you, and if you talk to your Felyne friend again, it'll inform you that the Gajalaka taught it how to speak Gastodonese, allowing your Palico to recruit Gastodons as a Tailraider. On top of all that, you can still claim even more rewards by talking to the folks at the Resource Center back in Astera! Generous! Finally, if you talk to the Lynian Researcher, he'll reward you further with another Silver Wyverian Print.
Setting Up Camp: Elder's Recess II
Return to the Elder's Recess and take one of the Gajalaka tunnels (either tunnel 4 or 5, which can be found in AREAS 8 and 7, respectively) to return to the Gajalaka camp in AREA 16. From the Gajalaka camp head south to reach a cavern overlooking a tunnel that leads to AREAS 12 and 14, which just so happens to be an ideal spot for a new camp. Merely entering the area should start the bounty Setting Up Camp: Elder's Recess II, which you can complete at your leisure. To do requires you to turn in 1,200 Research Points, a Dodogama Hide+ and two Windrake Hides.
Research Help: Bomb Beetle Capture
Make your way northeast from the Southern Camp and you'll find the Endemic Life Researcher in AREA 2. Talk to her and she'll request you capture her a Bomb Beetle so she can study their oddly suicidal habit of rolling around explosives. Continue northeast to the large, ridged tunnel, then follow it southeast down to AREA 6, where you should find a Bomb Beetle scurrying around. Just be wary of the Uragaan, as it likes to roll around these areas.
Research Help: Platinumfish Capture
From the Southern Camp make your way south, crawl through a hole, and look for the Piscine Researcher to find him fishing near a pool of water. Chat with him and he'll ask you to prove your mastery of the fishing arts by catching him a Platinumfish, which of course, can be found elsewhere. This is a pretty easy task, just fast travel to Eastern Camp and find a pool of water to the east which is swimming with fish... or rather, in which fish are swimming. You can find all kinds of fish in here, but the fish you're looking for is along the northern end of the pool. The Platinumfish isn't any harder to catch than, say, a Goldenfish, just don't tease your bait when it's in the water, and if no fish are interested, simply recast and try again.
To get this, you must have completed the Critical Bounty Cultural Exchange: Gajalaka Linguistics II and hence gained access to AREA 16. Ideally you'll have also completed the delivery request Setting Up Camp: Elder's Recess II. If you have it, fast travel to the Northwest Camp, or otherwise make your way on foot to AREA 16. Even if you have the Northwest Camp, you'll need to leave, make your way north through AREAS 14 and 15 and from there enter the Gajalaka camp... or use one of the Gajalaka tunnels to travel there. Either way, from the Gajalaka camp head south down a river until, near a short waterfall, you'll find an Amber Deposit. As usual for ultra-rare materials, this will only be here during an Upsurge: Amber Deposits event. Search the Amber Deposit to score some Twilight Stones.
Visit the Elder's Recess during an Upsurge: Beryl Deposits event and make your way to AREA 8. Head over to the western end of the area where several Gajalaka stand, near which is a cliff you can jump down to reach a crystal walkway below. Jump down, turn around, and loot the Beryl Deposit there to score some Noahstones.