On a second order eigenvalue problem and its application (Q2137836)

In this paper, authors consider a linear eigenvalue problem of a second order differential equation \[ d(\alpha) [D(x) \varphi_x]_{x} - \alpha [B(x)\varphi]_{x} + c(x)\varphi + \lambda \varphi =0, \quad x \in (0, L) \tag{1} \] with separate and general boundary conditions. Based on nice properties of the associated Fréchet operator, authors establish a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term under some assumptions. As an application, authors use this monotonicity result to study a class of competitive parabolic systems and observe the so-called competitive exclusion principle in a larger parameter region than several existing works.