This achievement was significantly easier than the first Jurassic Park level. You simply need to follow the instructions to capture dinosaurs, house them in an enclosure and then tranqualise and transport them off site.
When the mission starts try to build a control centre and response facility to the left and right of the arrival point as close as you can get with a backup generator at the top right corner behind the arrival point. You will then need a science centre which I built to the right of control centre.
Next you are tasked with capturing the Anklyosaurs which you must do by manually flying the helicopter from the response station. Before that build a pen just to the right of the science centre from the top of the map stretching as far right as you can and maybe a third of the way down the available map space. Be sure to leave enough room just beside the science centre for a viewing platform looking right towards the enclosure.
Next fly to where the compass at the top of the screen has the dinosaur icon, 2 hits on the Anklyosaurs with the tranquilliser will put it to sleep. Use the transport icon on the left to then collect it and put it in your ready and waiting pen. Be sure to have the ranger post in the enclosure and assign your ranger team for status check. Once you satisfy it's comfort needs you then need to put it to sleep again (can task this with the response helicopter rather than do it manually now) then transport it off the island.
After that it is a simple case of rinse and repeat with differing dinosaurs.
When you get to the Ceratosaurs (3rd) you will need to expand the pen a bit. I did this by building it a bit further left so it makes a sort of J shape. Only needs to go about half way across the map so it kind of loops under the science centre. Also this dinosaur will get sick so you need to build a medical facility which I built below the control centre to the left of the expanded pen. You will need to research this which you do at the science centre. You may need to hire a second scientist to have enough points available for the research. Once it's built, set the mobile unit to scan the dinosaur then it will need tranquillised and flown to the medical facility. You then need to assign scientists to treat it. I hired a second one to get enough points to heal it, not sure if everyone will need to do this.. Once it is all better simply transport it off site.
Next is the triceratops which you first need to drive a ranger truck out for and manually scan it with the binoculars. You simply need to get close enough to do so. After that use the helicopter to tranquilise and move it to the pen, satisfy it's needs and tranquilise again before transporting off site.
For the final lot of dinosaurs you will need to house 3 Brachiosaurus which require the rest of the available space for a second pen. You could possibly just house everything together but having 2 worked for me, I found it easier to manage needs having two pens rather than 1. Like with the first dinosaur you need to simply tranqualise them, transport to the pen, satisfy their needs and then transport off island. The Brachiosaursneed three shots to knock them out but they are hard to miss. The Parasaurolophus take 2 hits which is difficult because they run off at speed after the first hit. You just need to follow them and wait for them to slow down again. The Gallimimus just need 1 hot.
Each time you change the dinosaur in the pens throughout the mission you will need to change the setup to meet their needs. The list of what they need is below.
Ankylosaurus - Water / Ground Leaf / Groud Fruit
Stegosaurus - Water / Forest / Ground Fruit / Ground Fibre
Ceratosaurs - Carnivore Feeder / Forest / Water
Triceratops - Water / Forest / Ground Fibre
Brachiosaurs - Forest / Water / Tall Nut
Parasaurolophus - Water / Ground Nut
Gallimimus - Forest / Water / Ground Leaf
Once you satisfy the last lot, tranquilise and send off site the achievement will pop. If I've missed anything just let me know in the comments and I'll update.