From ODE to DDE∗

Delay Independent Stability of Linear Switching Systems with Time Delay

We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result o...

BIFURCATION ANALYSIS OF A DDE MODEL OF THE CORAL REEF

‎In this paper‎, ‎first we discuss a local stability analysis of model was introduced by P‎. ‎J‎. ‎Mumby et‎. ‎al‎. ‎(2007)‎, ‎with $frac{gM^{2}}{M+T}$ as the functional response term‎. ‎We conclude that the grazing intensity is the important parameter to control the existence or extinction of the coral reef‎. ‎Next‎, ‎we consider this model under the influence of the time delay as the bifurcat...

Convergence of the Variable-step Variable-order 3-stage Hermite–birkhoff Ode/dde Solver of Order 5 to 15

The ordinary and delay differential equations (ODEs/DDEs) solver HB515DDE is based on the discrete hybrid variable-step variable-order 3-stage Hermite–Birkhoff ODE solver of (consistency) order 5 to 15. The current version of the solver can handle ODEs and DDEs with statedependent, non-vanishing, small, vanishing and asymptotically vanishing delays. Delayed values are computed using Hermite int...

New ODE and DDE models for

Modeling biological oscillations: integration of short reaction pauses into a stationary model of a negative feedback loop generates sustained long oscillations

Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-DDE model, or introdu...