The paper, motivated in large part by the stability in a convection-like problem involving the gravity dependent motion of a suspension of swimming micro-organisms, is aimed to investigate the connection between the symmetric part of a linear operator and stability boundaries determined by the methods of linear theory and of nonlinear energy theory. It is proved that provided exchange of stabilities holds one can correlate linear and nonlinear stability. The method employs an idea of changing the energy norm by an appropriate spatial weight which renders the resulting linear spatial operator symmetric. As the proposed energy stability result relies on finding the maximum of a certain quotient sufficient conditions are given for the demonstration of existence of the maximum.