Qualitative Features of Delay Differential Equations

Ordinary and partial differential equations have long played an important role in bioscience, and they are considered to continue to serve as indispensable tools in future investigations as well. However, they frequently provide only a first approximation of the systems under consideration. More realistic models need to include some of the past states of these systems as well; that is, a real system needs to be modeled using differential equations with time-delays (or time-lags). Delay models formulated in mathematical biology include several types of functional differential equations, such as delay differential equations (DDEs), neutral delay differential equations (NDDEs), integro-differential equations, and retarded partial differential equations (RPDEs). Recently, stochastic delay differential equations (SDDEs) have attracted significant attention from researchers.