You're probably thinking of resin, and I don't think its something an amateur can do. Permeable, resin bound or resin bonded stones are often seen used around street tree planting these days, also used for pathways and other applications, but the resin and the stones are applied at the same time. Some information in the link below

http://www.resin-bound-and-bonded-surfacing.ronacrete.co.uk/

If you attempt a DIY job, make sure you use something that will be permeable for rainwater.

It's too late to do that now unless you want a potentially toxic mess on your hands. The solution is this ..... Remember how you leveled the gravel surface when you first laid it down? Well, do the same thing every few days or week. From the look of the size of your beds they are not large and should only take a few minutes to do. P.S. I like your dog:)