The reason why I ask is that I managed to get a five population large carnivore/fat tissue species and for at least five rounds I let it feed on my own cards that I had drawn during that round. [...] This gave me a huge advantage in a two-player game. Maybe it was just a beginners thing/luck. The other player never got a carnivore card so that he could reduce the population of my carnivore, but I guess this would have been the way to put a stop to this.
Yes, if the other player had drawn a carnivore card, your fat carnivore would have been in trouble.
You were certainly in a good position, but that just means your opponent's strategy would have to shift to "eat more food than you are eating" if they can't eat you. It seems like the amount of food you could get each round was essentially capped (any new cards you drew were going to your carnivore's food supply). There are other traits which if your opponent had drawn them could have helped them out, such as scavengers, long neck, and possibly fertile if you were not attacking anything slightly defended.
Tis the nature of the card game beast--it's hard to do things if you don't draw the right cards, but it makes each game uncertain/unpredictable.