Can someone please explain what "punctuated equilibrium" means exactly and how it ties with Charles Darwin’s theory of evolution. Beside "punctuated equilibrium", what are some specific contributions Dr.  Stephen Jay Gould has made to the study of evolution?

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Punctuated equilibrium is the idea that evolutionary changes can happen in fits and starts, rather than slowly and continuously.  The notion of "punctuated" can be thought of as a kind of metaphor, with punctuation being something that stops a process, just like a period stops a sentence.  Imagine birds, for example, that go through a period of drought.  Those that have a slight variation in their beaks that allow them to find food more easily will survive and pass on this trait. Those that have another kind of beak will not survive.  Thus, in one generation, a dramatic change occurs, and there might be no further changes for many generations, unless and until there are external forces to created another punctuated change.   In a continuous evolutionary situation, there would be tiny, incremental changes that would take many generations to "settle" into a useful trait. 

Another way to look at this is to think of that game in which you change one letter of a word, then the next person changes one more letter, and this continues until there is an entirely different word.  That would be continuous evolution.  However, if you picture a word in which several letters are changed to get a new word, that would be punctuated evolution. 

Darwin's theory of evolution is by no means disproved by the idea of punctuate evolution, in my opinion.  His theory, simply put, is that all living creatures have evolved over millions of years, in response to their environments, as opposed to having been put on this earth in their present forms by some divine creator.  Today, we have more information and better tools that confirm the idea that this sometimes happens more quickly and dramatically than Darwin realized. 

I have provided links for a good article on evolution and for a good article on Stephen Jay Gold. 

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