While there are many many variables that Amazon measures and calculates to decide which products show up in search results, the ultimate ranking factor for Amazon is sales.
Simply put, you need a certain amount of sales per day for a certain keyword to get to page 1 and stay there. Based on how competitive your niche is and the keywords you select, this could be a few sales per day for 5-7 days or hundreds of sales for 2-3 weeks.
Typically, a new listing is easier to rank as there are little to zero days of no sales. Why? Simply put, a certain metric that Amazon measures how many days your listing does not have any sales.
You can certainly rank a more established listing that has been around for a longer period of time, you just have to be more aggressive in your promotions to overcome those days of 0 sales.