Systems of Delay Differential Equations: Analysis of a model with feedback

Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in the {next} gland, which successively generates another hormone to stimulate the next one and so on. In the end, a final hormone is released which, by increasing its concentration, will inhibit the secretion of previous hormones that allowed the production process. This generates the decay of this hormone to a minimum threshold that re-activates the cycle again.\\ This behavior can be seen in other biochemical processes, such as enzymatic or bacterial models.\\ Topological degree is a useful tool to find stable equilibria in a wide variety of models with constant parameters and, furthermore, allows to deduce the existence of periodic solutions when the constant parameters are replaced by periodic functions.