Suppose there are i, j, and k individuals of the three species. Look at the differences i-j, i-k, and j-k, and in particular, look at these differences modulo 3 (i.e., look at their remainders when divided by 3).

Then, observe that no breeding event changes these numbers modulo 3. That is, when one of i, j, or k grows by 2, and the other two shrink by 1, the differences either stay the same or change by 3.