To determine the half-life of the compound, you are going to have to find out what happens to the compound. You need to know at least the major chemical pathway of decomposition, and it may be very different depending on whether it is in air, water, or soil. In particular, organisms living in those environments my regard tryptophan as a food source, as alluded to by Arkcon.
Once you know the chemical pathway of decomposition, you can analyze the kinetics of that pathway. Only then will you be able to calculate a half-life.
It is possible that somebody has already done this research, but a couple of quick Google searches haven't turned up that information (which I suspect you have also tried). The only kinetic work I have ever run across with tryptophan is in living organisms and has to do with metabolic pathways. That won't help you much for soil, air, and water decomposition.
Good luck