Problems with modelling using delay differential equations.

Анотація

In applications models described in the framework of delay differential equations (DDEs) are often used. The advantage of such approach is simplifying a description of complex natural phenomena which take some time. Using even one DDE with single delay one can reflect oscillatory dynamics typical for many biological systems. One ODE with two delays or two DDEs with single delay are sufficient to reflect stability switches with increasing delay. Simple DDEs models can also reflect chaotic dynamics. Although DDEs can be very useful in applications, they lead to much more complicated mathematical analysis than in ODEs case. DDEs define infinite dimensional semi-dynamical systems. Comparing to appropriate ODEs it should be noticed that DDEs not necessarily preserve non-negativity of solutions, it can be difficult to study global existence of solutions, and moreover global stability can be really hard to prove. Therefore, proposing the model based on DDEs one should be very careful and check at least basic properties to be sure that the model is properly defined.