by Tony Humphries, McGill
Coauthors: Sue Ann Campbell, Waterloo

Delay Differential Equations arise in many applications from pharmacokinetics to ecology to laser physics. While the theory of fixed discrete delay DDEs is now mature, equations arise from applications with distributed and/or state-dependent delays which fall outside the scope of much of this theory. This is a thriving field in Canada, and recent developments will be presented in this session.

Speakers:

Jacques Bélair, Université de Montréal
Pharmacokinetics modeling: when the delay is an effect (and vice-versa)

In modeling the time course of drug concentration after administration (pharmacokinetics), the simplest paradigms (instantaneous and homomogeneous mixing, for example) are oftentimes not entirely satisfied. This has lead classical phamacokineticians to the introduction of a so-called Effect compartment, which is, in effect (!!), a time delay in the kinetic description. We present a proper formulation of this trickery, including possible consequences of distributions of delays.

Pietro-Luciano Buono, UOIT
Bifurcation in symmetric delay-coupled rings of lasers

The Lang-Kobayashi (LK) equations are a model for semi-conductor lasers. We study delay-coupled systems of LK equations in a ring with Z_n and D_n symmetry. The basic solutions are Compound Laser Modes (CLMs) which are rotating waves and we classify them via their symmetry. We study steady-state and Hopf bifurcation from CLMs using DDE-Biftool. However, some results given by DDE-Biftool on the location of bifurcation points are not accurate and we use block diagonalization of the linearization using the isotypic decomposition of the space to find the correct location of bifurcation points. This is joint work with J. Collera.

Sue Ann Campbell, Waterloo
Centre Manifolds for Systems with Distributed Delays

Models for systems with distributed delays often include distributions that extend to infinite delays. While the theory of centre manifolds for systems with infinite delay is not complete, formal calculations can be carried out. We show that these formal calculations can be verified in certain special cases and consider some approximations for the centre manifold that can be computed without full knowledge of the distribution.

Jian Fang, York
Monotone traveling waves for delayed Lotka-Volterra competition systems

In this talk, we consider a delayed reaction diffusion Lotka-Volterra competition system which does not generate a monotone semiflow with respect to the standard ordering relation for competitive systems. We obtain a necessary and sufficient condition for the existence of traveling wave solutions connecting the extinction state to the coexistence state, and prove that such solutions are monotone and unique (up to translation). This talk is based on a joint work with Dr. Jianhong Wu.

Tony Humphries, McGill
Periodic Solutions of a Singularly Perturbed Differential Equation with two state-dependent delays

We consider a singularly perturbed delay differential equation (DDE) which has two linearly state-dependent delays, and show how to construct large amplitude periodic solutions in the singular limit using geometric arguments. Small amplitude solutions, near to the Hopf bifurcations, are constructed using Lindstedt series. Hence the bifurcation diagram is constructed in the singular limit. The bifurcation structures are shown to persist numerically close to the singular limit using DDEBiftool is used to demonstrate numerically that these bifurcation structures persist close to the singular limit.

Felicia Magpantay, York
Numerical simulations of an age-structured population model with a state-dependent threshold condition

We consider an age-structured population model, based on the Gurtin and MacCamy (1974) model, with a state-dependent threshold condition. This condition takes into account the effect of competition for resources on the maturation time of the population. Numerical methods for solving this equation are presented, as well as some interesting numerical simulations.

Gail Wolkowicz, McMaster
Effect of Distributed Delay on the Dynamics of a Predator-Prey Models

Rich dynamics have been demonstrated when a discrete delay is introduced in a simple predator-prey model to describe the time between first contact with the prey, and its eventual conversion to predator biomass. For example, Hopf bifurcations and a sequence of period doublings that appear to lead to chaotic dynamics are observed. A comparison will be made with the dynamics of the analogous model in the case of distributed delay.

Jianhong Wu, York
Global dynamics in reaction-diffusion systems with delayed unimodal nonlinearities

Delayed differential systems with unimodal nonlinearities arise from many applications and can exhibit complicated dynamical behaviors. The unique feature of the change of monotonicity of the nonlinearity however makes it possible to develop special type of comparison arguments to gain insights how complicated dynamics occurs. We demonstrate this with a scale reaction-diffusion equation with non-local delayed feedback.

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