Recent advances in theory and applications of functional differential equations
Org:
Victor Leblanc (Ottawa)
[
PDF]
 KEVIN CHURCH, McGill University
Computerassisted proof of Hopf bifurcation in functional differential equations of mixed type [PDF]

I will present a computational approach to Hopf bifurcation verification for functional differential equations of mixed type. The verification of a steady state, imaginary eigenvalues and their transversality amounts to a finitedimensional problem which we rigorously solve using a NewtonKantorovichtype theorem. To prove the imaginary eigenvalues are simple and that there is no resonance, we use some a priori estimates and rigorous contour integration of the characteristic equation to count all eigenvalues in a neighbourhood of the imaginary axis. As an application, we prove some results on periodic traveling waves in the Fisher equation with a nonlocal reaction term. This is joint with with JeanPhilippe Lessard.
 TERESA FARIA, University of Lisbon
Stability for nonautonomous linear delayed differential systems [PDF]

We study the stability of general nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of bounded diagonal delays are established. Our results encompass DDEs with discrete and distributed delays, and enhance some recent achievements in the literature.
 VICTOR LEBLANC, University of Ottawa
Degenerate Hopf Bifurcation in DDEs and Endemic Bubbles [PDF]

We consider 2parameter families of retarded functional differential equations (RFDE) which undergo Hopf bifurcation from an equilibrium, but for which the crossing condition of the Hopf theorem is violated (such a degeneracy is codimension 2). We classify the possible bifurcation diagrams in terms of the nonlinearities of the RFDE, and we apply the results to an SIS disease model incorporating delayed behavioral response.
 ANDRĂ‰ LONGTIN, University of Ottawa
Nonmonotonic complexity with increasing numbers of delays [PDF]

We investigate transitions to simple dynamics in firstorder nonlinear differential equations with multiple delays. Multiple delays can destabilize fixed points and promote highdimensional chaos, but many delays can also induce stabilization onto simpler dynamics. We focus on this behaviour as a function of the number of delays. Dynamical complexity is shown to depend on the precise distribution of delays. A narrow spacing between individual delays favours chaotic behaviour, while a lower density of delays enables stable periodic or fixed point behaviour. During complexity decrease, the number of roots of the characteristic equation around the fixed point that have a positive real part decreases. These roots behave in fact in a similar manner to the Lyapunov exponents and the KolmogorovSinai entropy for these multidelay systems, and can thus serve as a proxy for those dynamical invariants. Our results rely on a novel method to estimate the Lyapunov spectrum of multidelay nonlinear systems, as well as on permutation entropy computations. Surprisingly, complexity collapse upon adding more delays can occur abruptly through an inverse perioddoubling sequence. Our results shed light on the dynamical effects of the transition from discrete to continuous delay distributions. We also discuss the implications of our results for reservoir computing.
 MICHAEL MACKEY, McGill University
State dependent delays induce novel dynamics in gene regulatory systems [PDF]

This talk will review models for the bacterial regulation of gene expression and function for both repressible (negative feedback) and inducible (positive feedback) genes, and the nature of the nonlinearities involved. I argue that both the delays due to transcription of DNA to mRNA and translation of mRNA to produce protein are likely state dependent. The consequences of this turn out to be relatively astonishing in the sense that the state dependence of these delays can lead to completely new dynamical behaviors that are not present when the delays are constant. In both inducible and repressible systems the state dependence in the delays may lead to the appearance of more steady states as well as unexpected bifurcations not present when the delays are constant.
This is joint work with T. Gedon, A. Humphries, H.O Walther, and Z. Wang.
 FELICIA MAGPANTAY, Queen's University
LyapunovRazumikhin techniques for statedependent delay differential equations [PDF]

We present theorems for the Lyapunov and asymptotic stability of the steady state solutions to general statedependent delay differential equations (DDEs) using LyapunovRazumikhin methods. These theorems build upon the previous work of Hale and Verduyn Lunel (1993), and Barnea (1969) which were mainly aimed at equations with simpler delay terms (e.g. constant and timedependent delays), and are less applicable to statedependent DDEs such as the following model equation,
\[\dot u(t) =\mu u(t)+\sigma u(tacu(t)).\]
The stability region $\Sigma_\star$ of the zero solution to this model problem is known, and it is the same for both the constant delay ($c = 0$) and statedependent delay ($c \ne 0$) cases. Using our results we can prove the asymptotic stability of the zero solution to this model problem in parts of $\Sigma_\star$, considerably expanding upon the work of Barnea who proved Lyapunov stability for the simpler $\mu = c = 0$ constant delay case. Similar techniques are used to derive a condition for global asymptotic stability of the zero solution to the model problem, and bounds on periodic solutions when the zero solution is unstable. This is joint work with A.R. Humphries
 CONNELL MCCLUSKEY, Wilfrid Laurier University
Modelling the growth of variants [PDF]

There is a slow growth in the number of variants of concern for COVID19. We model this growth as proportional to the number of infected individuals worldwide. Once new variants appear, they contribute to the spread.
$\textrm{Let } M(t) \textrm{ be the number of variants, and let } i(t.m) \textrm{ be the number of individuals infected with variant } m \textrm{ at time }t$. Then
\begin{equation*}
\frac{dM}{dt}(t) = \int_{m=0}^{M(t)} p(m) \,i(t,m) \, dm,
\end{equation*}
$\textrm{where } p(m) \textrm{ is the rate at which variant } m \textrm{ slowly produces new variants.}$
$\textrm{What can we do with it? What impact do vaccines have on } M(t)$?
 GAIL WOLKOWICZ, McMaster University
A DecayConsistent Model of Population Growth and Competition with Delay [PDF]

We derive an alternative expression for a delayed logistic equation
in which the rate of change in the population involves a growth rate
that depends on the population density during an earlier time period.
In our formulation,
the delay in the growth term is consistent with
the rate of instantaneous decline in the population given by the model.
Our formulation is a modification of
[Arino et al., J. Theoret. Biol. 241(1):109119, 2006]
by taking the intraspecific competition between the adults and juveniles into account.
We provide a complete global analysis
showing that no sustained oscillations are possible.
A threshold giving the interface between extinction and survival
is determined in terms of parameters in the model.
Our approach for analyzing the global dynamics
incorporates the theory of chain transitive sets
and the comparison theorem for cooperative delay differential equations.
We extend our delayed logistic equation
to a system modeling the competition of two species.
For the competition model
we provide results on local stability, bifurcation diagrams, and adaptive dynamics.
Assuming that the species with shorter delay produces fewer newborns
than the species with longer delay,
we show that there is a critical value $\tau^*$
such that the evolutionary trend is to take the delay as close to $\tau^*$ as possible.
This is joint work with ChiuJu Lin and TingHao Hsu
 HUAIPING ZHU, York University
Models with delays for the transmission and control of COVID19 [PDF]

The ongoing global COVID19 epidemic poses a huge threat to human wellbeing and public health, waves of outbreaks continue to surge one year after it was declared as a global pandemic. There have been extensive modeling studies for the transmission which have been contributing greatly to inform decisionmaking. In this talk, I will discuss the role of delays in the modeling and dynamics for COVID19 and their application in informing decision making. It is wellknown that dynamical models with or without delays generate complex asymptotic dynamics through bifurcations. Usually there are two types of delays: one is due to the development or evolution of the virus, like the latency and incubation delays; the other is human but nonviral related, like delays in tracing and testing, quarantine and isolation, treatment due to lack of medical and health resources, or delays due to other social and behaviors in response to the control strategies. I will present examples of modeling with the two types of delays in control and mitigation of the epidemics of COVID19 with application to inform rapid decision making. Examples of Wuhan and other selected cities will be presented.
 XINGFU ZOU, University of Western Ontario
On a predatorprey system with digestion delay and antipredation strategy [PDF]

In this talk, I will present a predatorprey model incorporated with both cost and benefit from the prey's antipredation response, together with a time delay in the transfer of biomass from the prey to the predator after predation. The model is a system of delay differential equations (DDEs). By analyzing this nonlinear DDE system, we obtain some insights on how the antipredation response level (indirect effect) and the biomass transfer delay jointly affect the population dynamics; particularly we show how the nonlinearity in the predation term mediated by the fear effect affects the long term dynamics of the model system. These results seem to suggest a need to revisit some existing predatorprey models in the literature by incorporating the indirect effect and biomass transfer delay.
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