Stability of equilibria in quantitative genetic models based on modified-gradient systems - Benjamin J. Ridenhour; Jerry R. Ridenhour (Journal of Biological Dynamics, 2018)
Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system $ x^{′ }=P(t)\nabla f(x) $ which arise as critical points of f, under the assumption that $ P(t) $ is positive semi-definite. It is shown that the condition $ \int ^{∞ }λ _{1}(P(t))\ {\rm d}t=∞ $ , where $ λ _{1}(P(t)) $ is the smallest eigenvalue of $ P(t) $ , plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria.