Dynamical Analysis for the Hybrid Network Model of Delayed Predator-Prey Gompertz Systems with Impulsive Diffusion between Two Patches

In this paper, we consider a hybrid network model of delayed predator-prey Gompertz systems with impulsive diffusion between two patches, in which the patches represent nodes of the network such that the prey population interacts locally in each patch and diffusion occurs along the edges connecting the nodes. Using the discrete dynamical system determined by the stroboscopic map which has a globally stable positive fixed point, we obtain the global attractive condition of predator-extinction periodic solution for the network system. Furthermore, by employing the theory of delay functional and impulsive differential equation, we obtain sufficient condition with time delay for the permanence of the network.

Determination of an impulsive diffusion operator from interior spectral data

In the present work, the interior spectral data is used to investigate the inverse problem for a diffusion operator with an impulse on the half line. We show that the potential functions {q_{0}(x)} and {q_{1}(x)} can be uniquely established by taking a set of values of the eigenfunctions at some internal point and one spectrum.

A delayed predator–prey system with impulsive diffusion between two patches

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete. However, in the real world, it is often the case that diffusion occurs at certain moment every year, impulsive diffusion can provide a more suitable manner to model the actual dispersal (or migration) behaviors for many ecological species. In addition, it is generally recognized that some kinds of time delays are inevitable in population interactions. In view of these facts, a delayed predator–prey system with impulsive diffusion between two patches is proposed. By using comparison theorem of impulsive differential equation and some analysis techniques, criteria on the global attractivity of predator-extinction periodic solution are established, sufficient conditions for the permanence of system are obtained. Finally, numerical simulations are presented to illustrate our theoretical results.

Half inverse problem for the impulsive diffusion operator with discontinuous coefficient

The half inverse problem is to construct coefficients of the operator in a whole interval by using one spectrum and potential known in a semi interval. In this paper, by using the Hocstadt-Lieberman and Yang-Zettl?s methods we show that if p(x) and q(x) are known on the interval (?/2,?), then only one spectrum suffices to determine p (x),q(x) functions and ?,h coefficients on the interval (0,?) for impulsive diffusion operator with discontinuous coefficient.