71 citations · 328 across the 17 of their papers we have counts for
18 papers
Practicalities of State-Dependent and Threshold Delay Differential Equations
A. R. Humphries, A. S. Eremin, Z. Wang
Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show…
Seasonal Forcing Dominated Dynamics of a piecewise smooth Ghil-Zaliapin-Thompson ENSO model
Samuel Bolduc-St-Aubin, Antony R. Humphries
The Ghil-Zaliapin-Thompson (GZT) model, a scalar delay differential equation with periodic forcing and time-delayed feedback, captures key features of the El Nino-Southern Oscillat…
Dynamics of a state-dependent delay-differential equation
Tomas Gedeon, Antony R. Humphries, Michael C. Mackey +2
We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model.…
Bistability, Bifurcations and Chaos in the Mackey-Glass Equation
Valentin Duruisseaux, Antony R. Humphries
Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equati…
Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop
Antony R. Humphries, Bernd Krauskopf, Stefan Ruschel +1
We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduc…
Operon Dynamics with State Dependent Transcription and/or Translation Delays
Tomas Gedeon, Antony R. Humphries, Michael C. Mackey +3
Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we st…
Numerical methods and hypoexponential approximations for gamma distributed delay differential equations
Tyler Cassidy, Peter Gillich, Antony R. Humphries +1
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are…
A Recipe for State Dependent Distributed Delay Differential Equations
Tyler Cassidy, Morgan Craig, Antony R. Humphries
We use the McKendrick equation with variable ageing rate and randomly distributed maturation time to derive a state dependent distributed delay differential equation. We show that…
Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model
Daniel C. De Souza, Antony R. Humphries
We explore the bifurcations and dynamics of a scalar differential equation with a single constant delay which models the population of human hematopoietic stem cells in the bone ma…
Transit and lifespan in neutrophil production: implications for drug intervention
Daniel Camara De Souza, Morgan Craig, Tyler Cassidy +4
We compare and contrast the transit compartment ordinary differential equation modelling approach with distributed and discrete delay differential equation models. We focus on Quar…
Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations
F. M. G. Magpantay, A. R. Humphries
We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asy…
Normal and pathological dynamics of platelets in humans
Gabriel P. Langlois, Morgan Craig, Antony R. Humphries +6
We develop a comprehensive mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo…
Resonance phenomena in a scalar delay differential equation with two state-dependent delays
R. C. Calleja, A. R. Humphries, B. Krauskopf
We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and…
A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internalisation
Morgan Craig, Antony R Humphries, Michael C Mackey
We develop a physiological model of granulopoiesis which includes explicit modelling of the kinetics of the cytokine granulocyte colony-stimulating factor (G-CSF) incorporating bot…
Lyapunov-Razumikhin techniques for state-dependent delay differential equations
A. R. Humphries, F. M. G. Magpantay
We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin…
Periodic Solutions of a Singularly Perturbed Delay Differential Equation With Two State-Dependent Delays
A. R. Humphries, D. A. Bernucci, R. Calleja +2
Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain…
Synchronization in ecological systems by weak dispersal coupling with time delay
Emily Wall, Frederic Guichard, Antony R. Humphries
One of the most salient spatio-temporal patterns in population ecology is the synchronization of fluctuating local populations across vast spatial extent. Synchronization of abunda…
Computing Invariant Manifolds and Connecting Orbits in the Circular Restricted Three Body Problem
Renato C. Calleja, Eusebius J. Doedel, Antony R. Humphries +2
We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordi…