Relation between concentration of sugar in syrup VS its spoil rate
For a clear direct response for your question: yes, there is a correlation between sugar concentration and spoil rate. The correlation is actually between any kind of solute and spoil rate, so that's the reason for using salt and sugar as agents for preserving foods. 

The technical name is "water availability" and the maths behind that is "free water equivalent". That's because bacteria and other microorganisms need what we call "free water" to grow and reproduce - if the water is "bound" to other components like sugar or salt, they cannot be used for growing. 

The minimal concentration of sugar is 65% weight by weight if it is a simple solution of sugar in water (a.k.a. simple syrup)

If you want to use less sugar, you have to include other chemicals (called preservative agents) to prevent spoilage and bacterial growth. 

Here is the easiest to digest information I could find for how to do the calculation, without getting into many technical details on the chemical properties