Pullback permanence in a non-autonomous competitive Lotka--Volterra model
The goal of the paper is to study the long-time behaviour of the non-autonomous competitive system \[ \begin{aligned} u_t-\Delta u=u(\lambda -a(t)u-bv) &\quad\text{in } Q\times (s,+\infty),\\ v_t-\Delta v=v(\mu -cv-du) &\quad\text{in } Q\times (s,+\infty), \end{aligned} \] where \(Q\subset \mathbb{R}^N\) is a smooth bounded domain, \(b, c, d\) and \(a(t)\) are positive. Both the forwards and pullback behaviour of system are analysed. The notion of ''pullback permanence'' is introduced. The existence of a non-autonomous attractor and conditions for pullback permanence are estblished. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.
attractors for non-autonomous differential equations
scientific article; zbMATH DE number 1915659
scientific article