Here is the full list of all 28 The Islander achievements.
Reach Level 20 in a Single Run
Catch a total of 100 Parrots
Buy a total of 250 Animals
Catch a total of 100 Fish
Plant a total of 250 Trees
Plant a total of 100 Trees
Plant a total of 25 Trees
Catch a total of 100 Starfish
Plant a total of 250 Bushes
Buy a total of 100 Animals
Plant a total of 250 Crops
Cut down a total of 250 Trees
Break a total of 250 Rocks
Plant a total of 100 Bushes
Plant a total of 100 Crops
Buy a total of 25 Animals
Buy a total of 200 Cocobots
Have at least 1M Gold
Buy a total of 100 Land Expansions
Have at least 100K Gold
Reach Level 10 in a Single Run
Build a total of 750 Decorations
Build a total of 250 Decorations
Reach Level 5 in a Single Run
Build a total of 100 Decorations
Plant a total of 25 Bushes
Plant a total of 25 Crops
Have at least 1000 Gold