Well, you actually can replace "between 2000 and 2005" with "over the past five years" in the right context. The biggest thing you have to make sure of is that you are writing the sentence in 2005. I'll give you an example: "Between 2000 and 2005, there was a drop in the crime rate." "Over the past five years, there has been a drop in the crime rate." (let's say I wrote this sentence in 2005). I had to change a few words in my sentence to make sure that the prepositional phrase matched my verbs, but the two phrases mean ALMOST the exact same thing. I think the problem you may run into is that you are writing your sentence in 2013 (this year). If you say "over the last five years," then you have to be talking about 2008 to 2013, or maybe 2007 to 2012. I hope that helps!

As Christopher implies, 'past' means the most recent. But I would consider the past years to exclude this one.

"over the past five years" = 2009-2013