Kibble Quibble

We recently got a new bag of food for our dog Eros, and we try to taper him off the old food, since he has a sensitive stomach. As I was serving it up this morning, I got to thinking about how the mixture of foods changes depending on how long we spread the transition. If every day we feed him a total amount d, and we spread T days of old food across b days, we can write

where r is the rate we swap the foods. This is the series for the triangle numbers, so we can replace the sum and solve for the rate:

Using this, we can look at how the rate changes depending on the total amount, and the number of days we spread across:

Note that we require b > T, since otherwise we'd be giving more than a day's food in a day. Since b and T are both measured in days, I wondered if the ratio, representing how much we spread out the old food, had a clear relation to the rate of change in the mixture.

I expected that the points would all fall on a single curve, but there seems to be some variation depending on the specific values for b and T. 

Whenever I hear Eros's stomach making terrible noises, I'm reminded of a bit from Terry Pratchett's Guards! Guards! – "No wonder dragons were always ill. They relied on permanent stomach trouble for supplies of fuel. Most of their brain power was taken up with controlling the complexities of their digestion, which could distill flame-producing fuels from the most unlikely ingredients. They could even rearrange their internal plumbing overnight to deal with difficult processes. They lived on a chemical knife-edge the whole time. One misplaced hiccup and they were geography."