Periodic patterns in simple biological differential delay models

Periodic patterns in dynamical behaviours of biological models described by simple form differential delay equations are studied. Mathematical models are given by a class of scalar delay differential equations with a multiplicative time periodic mixed coefficient and a nonlinear delayed negative feedback. The dynamics is studied analytically with supportive numerical simulation and justification of the theoretical outcomes. The principal nonlinearity involving the state variable is of the negative feedback type, the periodic multiplicative coefficient can change its sign, leading to equations with mixed positive-negative feedback. The existence of slowly oscillating periodic solutions of two different types is established. The theoretical analysis and derivation are based on the reduction of dynamics in the delay equations to that of interval maps. The theoretical outcomes are verified and supported by comprehensive numerical computations. The differential delay equations considered are generalisations of some well-known autonomous models from biological applications.