This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U 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 Lotka-Volterra system" by V.W. Noonburg.
See also: http://homepages.math.uic.edu/~jan/Demo/noon5.html.
i1 : noon5(RR_53)
2 2 2 2 2 2 2 2
o1 = {x x + x x + x x + x x - 1.1x + 1, x x + x x + x x + x x -
1 2 1 3 1 4 1 5 1 1 2 2 3 2 4 2 5
------------------------------------------------------------------------
2 2 2 2 2 2 2
1.1x + 1, x x + x x + x x + x x - 1.1x + 1, x x + x x + x x +
2 1 3 2 3 3 4 3 5 3 1 4 2 4 3 4
------------------------------------------------------------------------
2 2 2 2 2
x x - 1.1x + 1, x x + x x + x x + x x - 1.1x + 1}
4 5 4 1 5 2 5 3 5 4 5 5
o1 : List
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The object noon5 is a method function.