$${\mathcal{A}}$$-Stability of Global Attractors of Competition Diffusion Systems

We study the structural stability of global attractors (A{\mathcal{A}}-stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally A{\mathcal{A}}-stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C ²-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.