Eventually, it will run or you will go to the horse's page that you caught. Or if you get to 75% or up (I believe) you can try to call the horse and the same options or the horse will ignore you.

There used to be Wild Essence but for some reason it was removed and I do not know when, so all you got to go on is the horse icon being Green, Yellow, Orange or Red.

Once you run into a wild, you have to work on their Bonding level and it can go up in random % intervals. I've been able to use Try to Call button, at 70% and sometimes not until 85%, give or take. You, essentially, need to use items or pet the horse, until you get the button Try to Call, pop up and even then sometimes the horse can ignore you but usually on 2nd or 3rd try you are able to get it to come to you. This is why it's best to stock up on lassos, whistles and treat items to get the horse to come to you as it is always a guessing game on whether or not the horse will run away or not.

Thank you both! Much appreciated! 

Also keep an eye on your energy bar.  Petting takes 10 energy, so that option will not be lit up if you don't have enough energy.

For what it's worth, I don't think the increase in bond itself is random (but the success of any action+ whether the horse runs after an action seems to be in part random). 

I've noticed that different horses seem to like different treats - it's easiest to see this with the crafted treats.

The plain crafted treats increase bond by a max of 10 per treat. The sweet ones (crafted treat x molasses) increase bond by a max of 20. This depends on how much the breed likes that treat. It always seems consistent to me whether the breed favors a certain treat (but I could be wrong).

If you have the items & bits, you'll possibly get the most reward from crafted treats than un-crafted treats (which only seem to give a bond increase of between 1 to 4).

The call button always appears for me when I reach 75% bond. You can catch a horse before this button appears by using the whistle or lasso - these both seem to have a higher chance of working on friendlier horses that have a higher bond (but you can still sometimes get lucky on other horses). 

I think DustyAngel is right about wild essence being indicated by the color of the icon. Treats, petting and the lasso can all reduce wild essence it seems. It can take a few successful actions before the icon changes color though.