Stability of equilibria in quantitative genetic models based on modified-gradient systems

Abstract

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system $ \mathbf {x}^{\prime }=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 ^{\infty }\lambda _{1}(P(t))\ {\rm d}t=\infty $ , where $ \lambda _{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.

Keywords

• dynamical systems
• modified-gradient system
• equilibria
• asymptotic stability
• basin of attraction