Existence and globally asymptotical stability of periodic solutions for two-species non-autonomous diffusion competition \(n\)-patch system with time delay and impulses (Q545574)

The authors consider the following nonlinear impulsive delay differential system \[ \dot{x}_i(t)=x_i(t)(r_i(t)-a_{ii}x_i(t)-a_{i,n+1}(t)y(t)) \] \[ +\sum_{j=1}^m D_{ij}(t)(x_j(t)-x_i(t)),\;1\leq i\leq n, \] \[ \dot{y}(t)=y(t)(r_{n+1}(t)-\sum_{j=1}^m a_{n+1,i}x_j(t)-a_{n+1,n+1}(t)y(t) \] \[ -\int_{-\tau}^0 K(t,s)y(t+s)\,ds), \] \[ \Delta x_i(t_k)=b_{ik}x_i(t_k), ~\Delta y(t_k)=b_{n+1,k}y(t_k). \] By using Mawhin's continuation theorem and a suitable Lyapunov functional, a set of sufficient conditions for the global asymptotical stability of positive periodic solutions are obtained.