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.