In this paper, a discrete version of continuous non-autonomous predator-prey model with infected prey is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and global asymptotical stability of positive periodic solution of difference equations in consideration are established. An example shows the feasibility of the main results.
This paper explores the problem of delay-independent and delay-dependent stability for a class of complex-valued neutral-type neural networks with time delays. Aiming at the neutral-type neural networks, an appropriate function is constructed to derive the existence of equilibrium point. On the basis of homeomorphism theory, Lyapunov functional method and linear matrix inequality techniques, several LMI-based sufficient conditions on the existence, uniqueness and global asymptotic stability of equilibrium point for complex-valued neutral-type neural networks are obtained. Finally, numerical examples are given to illustrate the feasibility and the effectiveness of the proposed theoretical results.
We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number R0 characterizes the existence of equilibria as well as the global dynamics. The disease free equilibrium always exists and is globally asymptotically stable if R0 < 1, while for R0 > 1 an endemic equilibrium occurs which is globally asymptotically stable. If the two regions are connected via unidirectional transportation, the disease free equilibrium always exists, but for R0 > 1 two endemic equilibria can appear. In this case, the regional reproduction numbers determine which one of the two is globally asymptotically stable. We describe how the time delay influences the dynamics of the system.