Global dynamics of evolution systems with asymptotic annihilation

This paper is devoted to the research on the global dynamics for a large class of evolution systems with uniform asymptotic annihilation. We first introduce the concept of the asymptotic spectral radius and study the associated principal eigenvalue problem. Then we establish the asymptotic annihilation and the threshold dynamics for such systems. Finally we apply the developed theory to investigate the global attractivity of a positive fixed point for an integro-difference equation and the propagation phenomenon for cooperative reaction-diffusion systems with a shifting habitat.