Recently literature on the qualitative theory of functional differential equations and difference equations was growing very fast. Oscillation, persistence, stability, periodicity and global attractivity theories for differential and difference equations as a part of the qualitative theory have been developing rapidly in the past thirty years.
This is due to the fact that there are many problems appearing in mathematical modeling in Biology, Ecology, Medicine, Economy and Physiology, as well as in several branches of engineering which are related to nonlinear differential and difference equations.
On the other hand, a major task of mathematics today is to harmonize the continuous and the discrete to include them in one comprehensive mathematics, and to eliminate obscurity from both. Stefan Hilger in his Ph. D. Thesis introduced the theory of time scales, which has recently received a lot of attention, to unify continuous and discrete analysis [S.Hilger, Analysis on measure chains- a unified approach to continuous and discrete calculus, Results Math. 18, (1990), 18-56]. Not only can this theory of the so-called ''dynamic equations'' unify the theories of differential equations and difference equations, but also extends these classical cases to cases '' in between'', e.g., to the so-called q-difference equations. A time scale T is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models which are discrete in season (and may follow a difference scheme with variable step-size or often modeled by continuous dynamic systems), die out, say in winter, while their eggs are incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping population.
Main topics in qualitative analysis:
In the following we state what we study in the qualitative theory of dynamic equations (differential and difference equations):
(1) Finding conditions for all solutions to be oscillatory.
(2) Finding conditions for existence of nonoscillatory solutions. One usually
reduces this problem to an operator equation in some space and looks for its
fixed point.
(3) Studying the classification of nonoscillatory solutions and existence
criteria of various nonoscillatory solutions.
(4) Establishing comparison theorems, by using the comparison theorem one can distinguish the oscillation for a quite large class of equations from a class of equations with known oscillation. Linearized oscillation criteria may belong to this kind of problems. (Nothing knows for Linearzied oscillation of nonlinear dynamic equations until now).
5) Finding conditions for existence and nonexistence of oscillatory solutions.
6) Studying the behavior of oscillatory solutions, such as estimating the distance between adjacent zeros of oscillatory solutions and studying the variation of amplitude.
7) Using the degree theory and continuation theorem to prove the existence of
periodic solutions.
8) Studying other features of solutions, which is related to the oscillation and periodicity, such as the persistence, permanence, stability, attractivity and boundedness.
9) Finding various applications of the qualitative theory in Biology, Physiology, Economy, Medicine and Engineering.
Research topics for next two years
(1) Qualitative properties of some linear and nonlinear dynamic equations, delay dynamic equations and neutral delay dynamic equations of different types on time scales (first, second and higher orders);
(2) the qualitative analysis of population cells model with impulsive perturbations;
(3) the qualitative analysis of discrete respiratory dynamics model;
(4) the oscillation and global attractivity of respiratory dynamics
model in time scales to unify the results of continuous and discrete cases;
(5) stability of some economical models;
(6) the interaction between team of predators and team of preys;
(7) oscillation and global attractivity of red cells blood model on time scale to unify the results of continuous and discrete cases;
(8) oscillation and global attractivity of Nichlson’s blowflies model on time scale scale to unify the results of continuous and discrete cases;
(9) stability of Malaria model with delays;
(10) complete the project of the book " Global dynamics of nonlinear delay population models".
