Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction–diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction–diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka–Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction–diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions