Evolving Populations, and Speciation

How does a population evolve?Does it work like this:A parent gives rise to a child who happens to be in posession of a beneficial mutation. This mutation proves to be so useful that the child manages to reproduce with no problem. If its children also posess the beneficial mutation, then that family line will soon dominate the population, until all members of the population posess the mutation.Or does it work like this:An environmental change occurs, 'encouraging' mutations across the population. The mutations might all be different, but all are beneficial as far as survival in the environment is concerned. Over time, the beneficial mutations are pooled together to create new species.(Please forgive my crude way of explaining things; I'm not familiar with scientific terminology, but I'm eager to learn!)

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