Well, I guess the 'odds' in 'overcome the odds' means kind of 'diffculties'. But I cannot find such kind of explaining in dictionary (oxford dict online) nor in "Practical Everyday English (by Steven Collins)". The closest explaining, seem to me, is 'The chances, probalility of winning, being successful'. But then it doesn't explain well for the following sentence (from "BBC Life Story")
Animals have just one goal at the end of this journey, to leave offspring, and everyone begins its life with an irrepressible instinct to survive and overcome the odds.
What does 'odds' mean here?