A three-dimensional Lotka–Volterra model is given by

˙ x = x(1−2x+y−5z), ˙ y = y(1−5x−2y−z), ˙z = z(1+x−3y−2z).

Prove that there is a critical point in the first quadrant at P( 1 14 , 3 14 , 3 14 ). Plot possible trajectories and show that there is a solution plane x + y + z = 1 2 .

Interpret the results in terms of species behavior.