Throughout the history of colonization, plants and animals from the Old World were often introduced to the New World, where many successfully established themselves.
How it should work - I envisage that a ship coming from Europe will have a random chance of getting a feral plant or animal stowaway and its release into the New World would only be announced when the ship arrives at a port. Likewise, the process needs to be repeated for islands not joined to the mainland. Otherwise, you could buy samples to introduce these species deliberately from ships or wagon trains. Once it is introduced into the New World, the area covered with the feral plant or animal is increases gradually, much like with forests in Sid Meier's Alpha Centauri: http://strategywiki.org/wiki/Sid_Meier's_Alpha_Centauri/Terraforming#Forest
Here is the list of feral plants and animals I propose (note how feral plants and animals native to the New World such as Water Hyacinth and Cane Toads are not included)
Name
Where it went feral
Effects in game
Peaches
Georgia
Broadleaf and Mixed Forest produce +1 food; cannot spread to rain forest, boreal forest, scrub forest, desert or tundra; non-wood producing biomes it spreads to produces +1 lumber
Rabbits
Australia
Plains, Savannah, Grassland and Prairie all produce +1 food due to rabbit meat; but Expert Farmer bonus is reduced due to rabbits eating crops; savannah and scrub forest turn into desert after 10 years of rabbit presence in those tiles
Pigs
Entire New World
One quarter of forest tiles with pigs in them have Game bonus but those with Game bonus originally lose their Game bonus once pigs arrive in those tiles
Kudzu
Southern USA
Forest tiles with kudzu die off in 10 years of kudzu presence in those tiles and turns into its cleared equivalent; cannot spread to rain forest or boreal forest; forest tiles with Kudzu in them produce half the quantity of lumber
Wild cattle
USA/Mexico border
Scrub forest, desert and savannah produce +1 food
Rats
Entire New World
Cities with efficiency penalties get efficiency penalty for food doubled, other cities get -1 food in total; automatically on ALL naval vessels leaving Europe and port settlements with rats
Zebra Mussels
Great Lakes and other cool temperate bodies of fresh water in North America
Transmitted by seagoing ships from Europe entering a lake; lake produces -1 fish
Common Carp
Entire New World
River bonus on food production eliminated, -1 food for swamp forest, wetland and marsh tiles
Chestnut Blight
US Eastern Seaboard
Broadleaf forest produces -1 food and -1 lumber
Sea Lampreys
Great Lakes and other cool temperate bodies of fresh water in North America
Transmitted by seagoing ships from Europe entering a lake; lake produces half quantity of fish
Himalayan Blackberry
North America
+1 food for Mountain, Hill, Boreal Forest, Mixed Forest and Conifer Forest Tiles but those same tiles get +1 movement penalty
Castor Oil Plant
Subtropical and Temperate North & South America
Road tiles with Castor Oil Plant present get -1 food, -1 tobacco, -1 cotton due to Castor Oil Plant spreading along roads
Acacia
South America
Tropical forest, Scrub Forest and Broadleaf Forest get +2 lumber, -2 food because Acacia is a fast growing tree with good wood but supports comparatively less animals
Cats
Entire New World
Rat penalty removed; Game bonus in forest tiles with cats disappears; cannot enter areas with feral pig or fox infestation except for European settlements; automatically on ALL naval vessels leaving Europe and port settlements with rats
Horses
Non forested biomes of the entire New World
All settlements built on tiles with horses start with 5 horses; can be released from settlements on your orders or when your citizens set them free in protest of taxes made by the king
African Land Snail
Tropical areas of New World
Non-plowed tiles in tropical areas produce -1 food
Red Deer
Cool temperate areas of North and South America
Tundra, marsh and grassland produce +1 food
Fox
Entire New World except tropical areas
Non-tropical tiles with foxes lose food bonus from Game, horses, red deer, rabbits and goats
Brown trout
Temperate fresh water bodies of North and South America
+1 food on river tiles in temperate tiles
Bamboo
Subtropical areas of North and South America
Tropical forest, Scrub Forest and Broadleaf Forest get +3 lumber, -3 food because Bamboos are fast growing plants with good lumber but supports much less animals
Goats
Entire New World
Eliminates Peaches, Kudzu, Acacia, Castor Oil Plant, Bamboo and Himalayan Blackberry; turns Scrub Forest tiles into desert as soon as it enters them; Hill tiles get -1 food
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I forgot to mention Africanised Killer Bees - These should double mean that any unit in Rain Forest, Swamp or Tropical Forest has a 1 in 20 chance of disappearing in that turn due to death from bee stings
Throughout the history of colonization, plants and animals from the Old World were often introduced to the New World, where many successfully established themselves.
How it should work - I envisage that a ship coming from Europe will have a random chance of getting a feral plant or animal stowaway and its release into the New World would only be announced when the ship arrives at a port. Likewise, the process needs to be repeated for islands not joined to the mainland. Otherwise, you could buy samples to introduce these species deliberately from ships or wagon trains. Once it is introduced into the New World, the area covered with the feral plant or animal is increases gradually, much like with forests in Sid Meier's Alpha Centauri:
http://strategywiki.org/wiki/Sid_Meier's_Alpha_Centauri/Terraforming#Forest
Here is the list of feral plants and animals I propose (note how feral plants and animals native to the New World such as Water Hyacinth and Cane Toads are not included)
I forgot to mention Africanised Killer Bees - These should double mean that any unit in Rain Forest, Swamp or Tropical Forest has a 1 in 20 chance of disappearing in that turn due to death from bee stings
See this video for an illustration:
https://www.youtube.com/watch?v=-1GadTfGFvU