What is the relationship between changes in a species envionment and the effectivesness of the species reproductive strategies

Adaption, during adaptation certain traits in a species can be passed on. When a change in environment is presented certain traits begin to develop when the species reproduce due to mutations in the DNA. The more the new trait due to mutation, is shown during reproduction the more it passes on from parent to offspring. EX.1: Polar bears, they originated from a brown bear but as the land began to get cold and snowy, certain mutations occured causing one offspring to be born with white fur. Once that bear began to reproduce the trait became more common, and due to the adaptation the polar bear could blend in to its surroundings more making it easier to hunt prey. Brown bears were not able to keep producing and living due to slowly dying off from starvation causing the trait of white fur to be more common. The species begin to mate with other species with similar traits to keep on surviving. I hope this answered something if not please message me.