That is a great question! In my experience how you answer this question will vary upon the species and the resource. More specifically, how easy is it to observe feeding and to monitor the resource in question. A key point here is that it is not only the resource abundance, but rate of renewal which is often key in situ. Moreover, the traits you observe will be very species specific. Perhaps density is the answer, or biomass, but factors like growth rate, shell thickness, or reproductive output could also be key. You will need to pick the best parameter for your species!
However, the setups that I have used and seen before consist of experimental trials that examine some response variable (survival, growth, reproduction, etc) of each species individually. This is often repeated at different levels of the resource to see at what values of the resource the individual species can continue to flourish. In reality this can easily be your control replicate!
The next step involves combining the two species at a level of resources considered limiting. I would actually observe the competition at different levels of the resource to not only see when it becomes limiting, but at what levels the two species can coexist. It is also useful to have different replicates with different densities of the 2 species. This will help you get a handle on the density dependent component of the competition. The obvious interaction between density and resource level could also be illuminating. By observing what happens to your response variable (survival, growth, etc) in the different treatments you should be able to get a good idea of the competition between the two species.
I believe the amount of matter (say dry matter) consumed will depend on the palatability of the material. This is specifically true for roughage material. Young material will be consumed more avidly than mature material. Therefore relative (total) intake and rate if intake could be a criteria for study.
The nutritive value of the material will also be a matter for concern. Cattle (ruminants in general) are selective when it comes to feeding. They first select the young and succulent material first and then to move to move mature material.
I was thinking of a similar answer as Mr. Jayasuriya.
If two or more species compete for the same resource, let us say type A leafs, time can be a limiting factor.
Each species has a basal metabolic rate. If you can control for BMR then you can measure time spent feeding on type A leafs. Obtain a control sample and then compare that with your observation.