This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i55250U CPU at 1.60GHz.
There were 233 solutions found in 0.582 seconds (with a Bezout bound of 243).
The coefficients have been chosen so that full permutation symmetry was obtained.
Reference: "A neural network modeled by an adaptive LotkaVolterra system" by V.W. Noonburg.
See also: http://homepages.math.uic.edu/~jan/Demo/noon5.html.

The object noon5 is a method function.