Periodic patterns in simple biological differential delay models

TL;DR

The paper investigates delay differential equations with $T$-periodic coefficients and mixed feedback, proving the existence of slowly oscillating periodic solutions that either share the coefficient period $T$ or double it to $2T$. By reducing the DDE dynamics to one-dimensional interval maps via piecewise-constant approximations of $f$ and the coefficients, it derives explicit affine maps whose fixed points correspond to periodic solutions and establishes stability criteria in terms of the map slope $m$. The main contributions include explicit parameter regimes for Type I ($T$) and Type II ($2T$) solutions, a rigorous smoothing argument showing persistence of these dynamics under continuous approximations, and a framework for extending the results to more general periodic inputs. The findings offer insight into how seasonal or circadian-like periodicities can induce robust periodic behaviours in biological delay systems and suggest directions for incorporating broader periodic structures and nonlinearities.

Abstract

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.

Figures (2)

• Figure 1: Slowly oscillating piecewise affine solution (Type I) on initial periodic interval $[0,T], T=p_1+p_2+p_3.$
• Figure 2: Slowly oscillating piecewise affine solution (Type II) on initial periodic interval $[0,T]$.