Continuous and Discrete Dynamical Systems: Comparisons, Interactions, Challenges

The talk includes the discussion on connections between continuous (differential equations) and discrete (difference equations) dynamical systems. The methods which first appeared for continuous models and later were successfully applied to discrete systems are illustrated on the example of exponential dichotomy results. It is well known that autonomous differential equations with the only positive equilibrium (which frequently occur in mathematical biology) usually have monotone solutions which tend to this equilibrium. The discrete counterparts of these models, however, can demonstrate instability and even transition to chaos.There is an attempt throughout the lecture to establish a bridge between two types of models. The results are illustrated by population ecology models. Differential equations of mathematical biology can incorporate harvesting which may be either of continuous (a term in the equation) or discrete (impulsive) nature; the relation of the two approaches is discussed. Finally, we overview a theory of dynamic equations which allows to consider differential and difference equations as special cases of some more general equation.