### Session S12 - Delay and functional differential equations and applications

## Talks

Monday, July 12, 11:40 ~ 12:15 UTC-3

## A Kupka-Smale Theorem for a Class of Delay-Differential Equations

### John Mallet-Paret

#### Brown University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider generic properties of scalar delay-differential equations of the form $\dot x(t)=f(x(t),x(t-1))$. In particular, we prove that for generic nonlinearities $f$ all equilibria and all periodic solutions are hyperbolic. We note that the corresponding result for equations of the form $\dot x(t)=f(x(t-1))$ remains open. We discuss the significance of the result as it relates to one-parameter families of equations $\dot x(t)=f(x(t),x(t-1),\mu)$ and global continuation of periodic orbits.

Monday, July 12, 12:15 ~ 12:50 UTC-3

## Linearized stability and instability for $\dot x(t) = g(x'_t, x_t)$

### Bernhard Lani-Wayda

#### Justus-Liebig-Universität Gießen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

In joint work in progress with Jaqueline Godoy Mesquita, we try to improve results on stability by linearization, and prove corresponding results in the unstable case.

Joint work with Jaqueline Godoy Mesquita (Universidade de Brasília).

Monday, July 12, 12:50 ~ 13:25 UTC-3

## On stability and asymptotics of Mackey-Glass equations with two delays and neutral systems

### Elena Braverman

#### University of Calgary, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the first part of our talk, we observe that introduction of two variable delays can change properties of the Mackey-Glass equation $$\dot{x}(t) = r(t) \left[ \frac{a x(h(t))}{1+x^{\nu}(g(t))} - x(t) \right], \quad a>1, ~\nu >0.$$ There may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain $a$ and $\nu$ disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.

In the second part of the talk, we analyze exponential stability and solution estimates for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$ are delayed arguments. Stability tests are applicable to a wide class of linear neutral systems with time-varying coefficients and delays. In addition, explicit exponential estimates for solutions of both homogeneous and non-homogeneous neutral systems are obtained for the first time. These inequalities are not just asymptotic estimates, they are valid on every finite segment and evaluate both short- and long-term behaviour of solutions.

Joint work with Leonid Berezansky (Ben Gurion University of the Negev, Israel).

Monday, July 12, 13:25 ~ 14:00 UTC-3

## On persistence of the Nicholson's equation with multiple delays and non-linear harvesting

### Melanie Bondorevsky

#### Universidad de Buenos Aires - IMAS (CONICET), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

A generalization of the Nicholson's equation is considered. We analyze conditions to guarantee uniform persistence for an $N$-dimensional generalization of the Nicholson's equation with several delays and non-linear harvesting terms. In addition, the existence of $T$-periodic solutions shall be proved and, if the conditions are reversed, we will show that zero is a global attractor.

Joint work with Pablo Amster (Universidad de Buenos Aires - IMAS (CONICET)).

Monday, July 12, 14:15 ~ 14:50 UTC-3

## On almost periodic solutions for a model of hematopoiesis with an oscillatory circulation loss rate

### Rocio Celeste Balderrama

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The following nonlinear autonomous delay differential equation was proposed by Mackey and Glass to describe the regulation of hematopoiesis, namely, \begin{equation} \label{MG0} \frac{dP(t)}{dt}=\frac{\lambda \theta^{n}P(t-\tau)}{\theta^n + P^{n}(t-\tau)}-\gamma P(t)\, \hspace{1cm} \hbox{ where } \lambda, \theta, n, \gamma, \tau\in(0,+\infty).\quad (1) \end{equation} Here $P(t)$ is the concentration of cells in the circulating blood and the flux function $f(v)=\frac{\lambda \theta^{n}v }{\theta^n + v^{n}}$ of cells into the blood stream from the stem cell compartment depends on the cell concentration at an earlier time. The delay $\tau$ describes the time between the start of cellular production in the bone marrow and their maturation for release in circulating bloodstream. It is assumed that the cells are lost at a rate proportional to their concentration, namely $\gamma P(t)$.

In the real-world phenomena, the environment varies with time. Thus, the following nonautonomous nonlinear delay differential equation with time-varying coefficients and delays, and oscillatory circulation loss rate is a natural extensions of (1)

\begin{equation} \label{MG} x'(t) = \sum_{k=1}^M \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}-b(t)x(t), \quad (2) \end{equation} where $m\geq 0$, $n, \lambda_k>0$, $b(t),r_k, \tau_k$ are positive almost periodic functions for $k= 1, 2, \ldots, M$ and $b(t)$ is oscillating. The function $ \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}$ is the flux of cells into the blood stream from the $i-th$ stem cell compartment.

In this work, we establish and prove a fixed point theorem from which some sufficient conditions are deduced on the existence of positive almost periodic solutions for (2). Some particular conditions under the nonlinearity of the equation have been previously considered by authors as fundamental assumption for the study of almost periodic solutions of the model. The aim of this work is to establish results without such assumption. Some examples are given to illustrate our results.

Monday, July 12, 14:50 ~ 15:25 UTC-3

## $L^p$-Solutions for Kelvin-Voigt Fluids with Unbounded Memory Terms

### Pedro Danizete Damazio

#### Federal University of Paraná, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work we consider systems of Kevin-Voigt type with distributed memories in the nonlinear and force terms. By setting the problem in the context of the analytic semigroups in Banach spaces, results on existence, uniqueness and regularity of weak, mild and classical solutions are established.

Joint work with Ana Leonor Silvestre (Instituto Superior Técnico - Lisboa, Portugal) and Patrícia Aparecida Manholi (Universidade Tecnológica Federal do Paraná - Curitiba, Brazil).

Monday, July 12, 15:25 ~ 16:00 UTC-3

## The multiplicity-induced-dominancy property for scalar delay-differential equations

### Guilherme Mazanti

#### Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

Even in simple situations with time-delays such as that of linear equations with constant coefficients and constant delays, the analysis of the asymptotic behavior of a time-delay system can be a challenging question. From a spectral point of view, this asymptotic behavior can be studied through the spectrum of the system, which is the set of roots of a quasipolynomial, i.e., a function which can be written as a finite sum of products of polynomials and exponentials. Contrarily to the case of polynomials, for which Routh–Hurwitz criterion allows for handy characterizations of the location of their roots in terms of the coefficients of the system, quasipolynomials have infinitely many roots and there is no explicit link in general between their location and the coefficients of the system.

Some recent works have highlighted an interesting property of time-delay systems, called multiplicity-induced-dominancy (MID): for some families of time-delay systems, a spectral value of maximal multiplicity is necessarily the rightmost spectral value in the complex plane, and hence determines the asymptotic behavior of the system. Since then, an important research effort was made to identify families of time-delay systems satisfying the MID property.

After an introductory discussion on the spectral analysis of time-delay systems, this talk will present the MID property and some families of systems for which it is known to hold. We shall present the most common techniques used to prove the MID property, based on the argument principle, factorizations of quasipolynomials in terms of confluent hypergeometric functions, and the analysis of crossing imaginary roots. We will also illustrate its application to the stabilization of control systems and present the main perspectives of this ongoing line of research.

Joint work with Amina Benarab (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Catherine Bonnet (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Islam Boussaada (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Yacine Chitour (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Sébastien Fueyo (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Silviu-Iulian Niculescu (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Karim Trabelsi (IPSA, France) and Tomáš Vyhlídal (Czech Technical University in Prague, Czechia).

Tuesday, July 13, 11:05 ~ 11:40 UTC-3

## Stochastic age-structured population models derived from the McKendrick-von Foerster equaton

### Arkadi Ponossov

#### Norwegian University of Life Sciences, Norway - This email address is being protected from spambots. You need JavaScript enabled to view it.

A widespread method of producing a stochastic population model consists in appending an additive stochastic noise to the model’s deterministic counterpart. It is shown in the talk that this straightforward construction may result in a wrong equation, at least in the case of age-structured population models derived from the McKendrick-von Foerster equaton (MFE), the biological “conservation law”. This well-elaborated transformation technique produces many popular deterministic differential equations with time-delay such as Nicholson’s blowflies model, the recruitment-delayed model etc.

If one takes MFE with a stochastically perturbed mortality rate as a starting point and use a similar transformation technique, then, as it is demonstrated in the talk, one obtains stochastic versions of the above population models, which are very different from those one can construct by simply appending an additive stochasticity to deterministic equations. In particular, it is shown that stochastic Nicholson’s blowflies model should contain both additive and multiplicative stochastic noises.

Joint work with Lev Idels (Vancouver Island University, Canada) and Ramazan Kadiev (Dagestan State University, Russian Federation).

Tuesday, July 13, 11:40 ~ 12:15 UTC-3

## On solution manifolds

### Hans-Otto Walther

#### Justus-Liebig-Universitaet Giessen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

Differential equations with state-dependent delays which generalize the scalar example \[ x'(t)=g(x(t),x(t-d(x_t))) \] where $g:\mathbb{R}^2\to\mathbb{R}$ and $d:C([-r,0],\mathbb{R})\to[0,r]$ are continuously differentiable, and with $x_t:[-r,0]\to\mathbb{R}$ given by $x_t(s)=x(t+s)$, define semiflows of differentiable solution operators on an associated submanifold of the state space $C^1=C^1([-r,0],\mathbb{R}^n)$. When written in the general form \[ x'(t)=f(x_t) \] with a map $f:C^1\supset U\to\mathbb{R}^n$ then the associated manifold is \[ X_f=\{\phi\in U:\phi'(0)=f(\phi)\}. \] We obtain results on the nature of $X_f$.

1. If all delays in the system are bounded away from zero then a projection $C^1\to C^1$ onto the subspace \[ H=\{\phi\in C^1:\phi'(0)=0\}=X_0 \] defines a diffeomorphism of $X_f$ onto an open subset of $H$. In other words, $X_f$ is a graph over $H$.

2. There exist $g$ and $d$ with $d(\phi)>0$ everywhere and $\inf\,d=0$ so that the manifold $X_f$ associated with the scalar example above does not admit any graph representation.

3. If all delays in the system are strictly positive (but not necessarily bounded away from zero) then $X_f$ has an ''almost graph representation`` which implies that it is diffeomorphic to an open subset of $H$.

Tuesday, July 13, 12:15 ~ 12:50 UTC-3

## Laguerre-domain modeling and identification of linear discrete-time delay systems

### Rosane USHIROBIRA

#### Inria, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we propose a closed-form Laguerre-domain representation of discrete linear time-invariant systems with constant input time delay. This representation is shown to be useful in system identification setups involving square-summable signals, usually appearing in biomedical applications, and where experimental protocols do not allow for the persistent excitation of the system dynamics. We illustrate the utility of our proposed system representation on a problem of drug kinetics estimation from clinical data.

Joint work with Viktor Bro (Uppsala University, Sweden) and Alexander Medvedev (Uppsala University, Sweden).

Tuesday, July 13, 12:50 ~ 13:25 UTC-3

## Random recurrent neural networks with delays

### Xiaoying Han

#### Auburn University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will introduce an infinite lattice model of recurrent neural networks with random connection strengths among nodes, and various types of temporal delays. The existence and structure of random pullback attractors will be discussed.

Joint work with Peter Kloeden( Tuebingen University, Germany), Yejuan Wang (Lanzhou University, China) and Meiyu Sui (Hebei Normal University, China).

Tuesday, July 13, 13:40 ~ 14:15 UTC-3

## A delay differential equation with an impulsive self-support condition

### Márcia Federson

#### Universidade de São Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We investigate an autonomous impulsive system proposed as a model of drugs absorption and consisting of a linear differential delay equation and an impulsive self-support condition. A representation of the general solution in terms of the fundamental solution is obtained. We also show that the impulsive self-support condition generates periodic auto-oscillations given by fixed points of a first return map.

Joint work with Istvan Gyori (University of Pannonia, Hungary), Jaqueline G. Mesquita (Universidade de Brasília, Brazil) and Plácido Táboas (Universidade de São Paulo, Brazil).

Tuesday, July 13, 14:15 ~ 14:50 UTC-3

## Multiple periodic solutions for dynamic Liénard equations with delay and singular $\varphi$-laplacian of relativistic type

### Mariel Paula Kuna

#### Departamento de Matemática, FCEyN, Universidad de Buenos Aires e IMAS-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work, we study the existence and multiplicity of $T$-periodic solutions $x:\mathbb{T} \rightarrow \mathbb{R}$ to the following equation with delay on time scales \[(\varphi(x^{\Delta}(t)))^{\Delta} +h(x(t))x^{\Delta}(t)+ g(x(t-r)) =p(t) \ \ \ t\in \mathbb{T},\] where ${\mathbb{T}}$ is an arbitrary $T$-periodic nonempty closed subset of $\mathbb{R}$ (\textit{time scale}), $\varphi:(-a,a)\rightarrow \mathbb{R}$ is an increasing homeomorphism with $0

\smallskip

Under appropriate assumptions, multiple solutions are obtained as fixed points of an operator arising on a nonlinear Lyapunov-Schmidt decomposition. \smallskip

A special case of interest is the sunflower equation with relativistic effects on time scales, namely \[ \left( \frac {x^\Delta(t)} {\sqrt {1- \frac{x^\Delta(t)^2}{c^{2}} }}\right)^\Delta+ ax^\Delta(t) + b \sin( x(t-r)) = p(t).\] We prove that if $T$ is small, then the equation has a $T$-periodic solution. The results improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$ and without delay. The bound shall be expressed in terms of $k(\mathbb T)$, the optimal constant of the Sobolev inequality $$\|x-\overline x\|_{\infty}\leq k\|x^\Delta\|_{\infty}, \;\; x\in C^1_T.$$

\noindent {\bf References}

\medskip

\noindent [1] P. Amster, M. P. Kuna and D. P. Santos, {\em Existence and multiplicity of periodic solutions for dynamic equations with delay and singular $\varphi$-Laplacian of relativistic type}, submitted.

\medskip

\noindent [2] P. Amster, M. P. Kuna and D. P. Santos, {\em On the solvability of the periodically forced relativistic pendulum equation on time scales}, Electron. J. Qual. Theory Differ. Equ. 2020, No. 62, 1-11.

Joint work with Pablo Amster (Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS-CONICET, Argentina) and Dionicio Pastor Santos (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina).

Tuesday, July 13, 14:50 ~ 15:25 UTC-3

## Solutions for functional Volterra--Stieltjes integral equations

### Anna Carolina Lafetá

#### Universidade de Brasília, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work, we study integral functional differential equations of Volterra--Stieltjes type:

$(1)\qquad\left\{\begin{array}{lll}x(t) &= & \phi(0) + \displaystyle \int_{t_0}^t a(t,s)f(x_s, s)\mbox{d}g(s), \qquad t\geq t_0,\\ x_{t_0 } & = & \phi,\end{array} \right.$

where $t_0 \in \mathbb{R}$, $\phi \in G([-r,0],\mathbb{R}^n)$, $f:G([-r,0],\mathbb{R}^n)\times [t_0,+\infty)\to \mathbb{R}^n$, $a:[t_0,+\infty)^2 \to \mathbb{R}$, $g:[t_0,+\infty) \to \mathbb{R}$, $x_s:[-r,0] \to \mathbb{R}^n$ is defined by $x_s(\theta) = x(s + \theta)$ and the integral on the right hand side fo the equality in the sense of Henstock--Kurzweil--Stieltjes.

We present some conditions on the functions $a$ and $g$ and also some conditions with respect the integral $\displaystyle \int^{\tau_2}_{\tau_1}b(t,s) f(x_s,s)\mathrm{d}g(s)$, when $b:[t_0,+\infty)^2 \to \mathbb{R}$ is a regulated function and $t_0 \leq \tau_1 \leq \tau_2 \leq t_0 + \sigma

These conditions will guarantee the existence and uniqueness of local and maximal solutions for equation (1).

Moreover, we present correspondences between (1) and functional Volterra integral equations with impulses and Volterra delta--integral equations on time scales.

References

[1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.

[2] M. Federson, R. Grau and J. G. Mesquita, Prolongation of solutions of measure differential equations and dynamic equations on time scales, Mathematische Nachrichten, 292(1), 22-55, 2019.

[3] M. Federson and Š. Schwabik, Generalized ODEs approach to impulsive retarded differential equations. Diff integral equations, 19(11) (2006) 1201-1234.

[4] D. Fraňková, Regulated functions, Mathematica Bohemica. 116 (1) (1991), 20-59.

[5] R. Grau, A. C. Lafetá and J. G. Mesquita, Existence and uniqueness of local and maximal solutions for functional Volterra Stieltjes integral equations and applications, submitted.

[6] G. Gripenberg, S.-O. Londen, O. Steffans, Volterra Integral and Functional Equations, in: Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990.

[7] R. Henstock, A Riemann-type integral of Lebesgue power. Canad. J. Math. 20, (1968) 79--87.

[8] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Series in Real Anal., vol. 5, 1992.

[9] A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl. 385 (2012), 534--550.

Joint work with Jaqueline Godoy Mesquita (Universidade de Brasília, Brasília, Brasil) and Rogélio Grau (Universidad del Norte, Barranquilla, Colombia).

Tuesday, July 13, 15:25 ~ 16:00 UTC-3

## Analysis of stability in neutral delay differential equations through different approaches

### Griselda Rut Itovich

#### Universidad Nacional de Río Negro, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

This work is about the analysis of equilibrium stability in certain neutral delay differential equations (ndde), which include several bifurcation parameters. In the considered cases, the associated characteristic equation is an exponential polynomial one with a principal term, so some Pontryagin results [1,2] are a convenient tool to locate its roots. Then, it is possible to set up certain regions in the parameter space where asymptotic stability can be guaranteed. Besides, the outcomes are all in agreement with those coming from a sophisticated application of the Nyquist stability criterion (based on the Cauchy's argument principle) and also with others found in the literature, related with some real systems modelled by ndde.

References

[1] Bellman R. and Cooke, K. Differential-Difference Equations, Academic Press, New York, 1963.

[2] Pontryagin, L. S. On the zeros of some elementary transcendental functions, American Math. Society Translations 21 (1), pp. 95-110, 1955.

Joint work with Franco Sebastián Gentile (Universidad Nacional del Sur, Argentina) and Jorge Luis Moiola (Universidad Nacional del Sur, Argentina).

Wednesday, July 21, 16:00 ~ 16:35 UTC-3

## Stability for nonautonomous linear differential equations with infinite delay

### Teresa Faria

#### Universidade de Lisboa, Portugal - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the asymptotic and the exponential asymptotic stability of general nonautonomous linear differential systems 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 improve some recent achievements in the literature.

[1] T. Faria, Stability for nonautonomous linear differential systems with infinite delay, J. Dyn. Diff. Equat. (2020). https://doi.org/10.1007/s10884-020-09873-0

Wednesday, July 21, 16:35 ~ 17:10 UTC-3

## Chaotic behavior in a functional differential equation

### Sergei Trofimchuk

#### Universidad de Talca , Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Differential equations with maxima can be regarded as a special subclass of differential equations with state-dependent lags. This kind of delay equations and related differential inequalities are known mainly due to the Halanay inequality and Myshkis-Yorke 3/2-stability theorems [1,2,3]. These equations also appear in a natural way in applications coming from the real world: one of them will be analysed in our talk. For this particular application, we prove the existence of solutions with complicated behaviour and discuss their relevance to the original real world model.

References:

1. N. Bantsur and E. Trofimchuk, Existence and stability of the periodic and almost periodic solutions of quasilinear systems with maxima, Ukrainian Math. J., 50 (1998), 847-856.

2. A. Ivanov, E. Liz and S. Trofimchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Mathematical Journal, 54 (2002) 277--295.

3. M. Pinto and S. Trofimchuk, Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima, Proc. Roy. Soc. Edinburgh Sect. A, {130} (2000), 1103-1118.

Joint work with Eduardo Liz (Universidade de Vigo, Spain) and Elena Trofimchuk (Igor Sikorsky Kyiv Polytechnic Institute, Ukraine).

Wednesday, July 21, 17:10 ~ 17:45 UTC-3

## On local stability of stochastic delay nonlinear discrete systems with state-dependent noise

### Alexandra Rodkina

#### The University of the West Indies, Jamaica - This email address is being protected from spambots. You need JavaScript enabled to view it.

We examine the local stability of solutions of a delay stochastic nonlinear difference equation with deterministic and state-dependent Gaussian perturbations. We apply the degenerate Lyapunov-Krasovskii functional technique and construct a sequence of events, each term of which is defined by a bound on a normally distributed random variable. Local stability holds on the intersection of these events, which has probability at least $1-\gamma$, $\gamma\in (0,1)$. This probability can be made arbitrarily high by choosing the initial value sufficiently small. We also present a generalization to systems where a condition for stability is expressed in terms of the diagonal part of the unperturbed system, and computer simulations which illustrate our results.

Joint work with Josef Diblik (Brno University of Technology, Czech Republic) and Zdenek Smarda (Brno University of Technology, Czech Republic).

Wednesday, July 21, 17:45 ~ 18:20 UTC-3

## Periodic positive solutions of superlinear delay equations via topological degree

### Pierluigi Benevieri

#### Universidade de São Paulo, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In recent years, some papers by G. Feltrin and F. Zanolin have been devoted to the study of existence and multiplicity of periodic solutions to nonlinear differential equations of the form \[ u''=f(t,u,u'), \] with periodic or Newmann boundary conditions and suitable assumptions on $f$. In a recent joint work with P. Amster and J. Haddad, that I present in the talk, existence and multiplicity of periodic solutions is proven for a class of delay equations of the type \[ u''(t)=f(t,u(t),u(t-\tau),u'(t)). \] The approach is, as in the work of Feltrin and Zanolin, topological and based of the coincidence degree introduced by J. Mawhin.

Joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal do Minas Gerais, Brasil).

Wednesday, July 21, 18:35 ~ 19:10 UTC-3

## Asymptotic behaviour of the energy to the viscoelastic wave equation with localized memory

### Valéria Domingos Cavalcanti

#### State University of Maringá, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We are concerned with the well-posedness of solutions as well as the asymptotic behaviour of the energy related to the viscoelastic wave equation with localized memory with past history and supercritical source and damping terms, posed on a bounded domain in the three-dimensional euclidean space.

Joint work with Marcelo M. Cavalcanti (State University of Maringá, Brazil), Talita D. Marchiori (State University of Maringá, Brazil) and Claudete M. Webler (State University of Maringá, Brazil)..

Wednesday, July 21, 19:10 ~ 19:45 UTC-3

## Global stability and periodic solutions in delayed chemostat models

### Gonzalo Robledo Veloso

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

The chemostat is a continuous bioreactor, where a set of microbial species is cultivated in a homogeneous liquid medium and whose growth is due to the consumption of a limiting nutrient that is pumped into it continuously. The evolution of the variables (nutrient and microbial species) is described by differential equations. However, if we consider the time interval elapsed between the consumption of the nutrient and its metabolization, the corresponding models are described by differential delay equations (DDE).

In homogeneous conditions, the asymptotic behavior of the chemostat equations is described by the competitive exclusion principle, namely, the microbial species that need the minimal amount of nutrients to have a positive growth will be the unique survivor, while the other species will be driven to extinction. In this context, a great amount of research and modeling effort has been carried out in order to avoid competitive exclusion and in this talk we will revisit recent approaches.

First of all, we will study a chain of two chemostats described by an autonomous system of DDE, where a chemostat with two competitors receives a continuous feeding from a second chemostat cultivating only the less advantaged competitor. The coexistence of the competitors can be achieved as a positive equilibrium, whose global asymptotic stability is deduced by using a Lyapunov Krasovskii approach.

Secondly, we will study two $\omega$--periodically perturbed models with one species and prove the existence of an $\omega$--periodic solution and its uniqueness for small delays. The techniques used are a classical coincidence degree theorem and Poincaré map techniques tailored to the DDE framework. In spite of the importance of the result in the field bioprocesses, we also highlight that the adaptation of Poincaré map techniques to DDE equations is interesting from a mathematical point of view.

Amster P, Robledo G, Sepulveda D, Dynamics of a chemostat with periodic nutrient supply and delay in the growth. Nonlinearity 33 (2020), no. 11, 5839-–5860.

Amster P, Robledo G, Sepulveda D, Existence of ω-periodic solutions for a delayed chemostat with periodic inputs. Nonlinear Anal. Real World Appl. 55 (2020), 103134.

Mazenc F, Niculescu SI, Robledo G. Stability analysis of mathematical model of competition in a chain of chemostats in series with delay. Appl. Math. Model. 76 (2019), 311-–329.

Joint work with Frédéric Mazenc (L2S - CNRS - CentraleSupelec, France), Pablo Amster (Universidad de Buenos Aires, Argentina) and Daniel Sepúlveda (Universidad Tecnológica Metropolitana, Chile).

Wednesday, July 21, 19:45 ~ 20:20 UTC-3

## A Delay Model for Persistent Viral Infections in Replicating Cells

### Gail Wolkowicz

#### McMaster University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Persistently infecting viruses remain within infected cells for a prolonged period of time without killing the cells and can reproduce via budding virus particles or passing on to daughter cells after division. The ability for populations of infected cells to be long-lived and replicate viral progeny through cell division may be critical for virus survival in examples such as HIV latent reservoirs, tumor oncolytic virotherapy, and non-virulent phages in microbial hosts. We consider a model for persistent viral infection within a replicating cell population with time delay in eclipse stage prior to infected cell replicative form. We obtain reproduction numbers that provide criteria for the existence and stability of the equilibria of the system and provide bifurcation diagrams illustrating \textit{transcritical (backward and forward), saddle-node, and Hopf} bifurcations, and provide evidence of {\it homoclinic bifurcations} and a \textit{Bogdanov-Takens bifurcation}. We investigate the possibility of long term survival of the infection (represented by chronically infected cells and free virus) in the cell population by using the mathematical concept of \textit{robust uniform persistence}. Using numerical continuation software with parameter values estimated from phage-microbe systems, we obtain two parameter bifurcation diagrams that divide parameter space into regions with different dynamical outcomes. We thus investigate how varying different parameters, including how the time spent in the eclipse phase, can influence whether or not the virus survives.

Joint work with Hayriye Gulbudak (University of Louisiana at Lafayette) and Paul L. Salceanu (University of Louisiana at Lafayette).

Wednesday, July 21, 20:20 ~ 20:55 UTC-3

## Dynamics of a Lamé system featuring damping-vs-delay

### To Fu Ma

#### University of Brasília, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we discuss the long-time dynamics of weakly dissipative elasticity systems delay effects on the velocity. This damping-vs-delay feature was firstly considered by Nicaise and Pignotti (2006) in the context of interior and boundary controllability for wave equations. Our objective is to establish the existence of a smooth finite dimensional global attractor, allowing nonlinearities of critical growth. The main difficulty is finding necessary arguments to show that the system is quasi-stable in the sense of Chueshov and Lasiecka.

References:

[1] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006) 1561-1585.

[2] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.

[3] T. F. Ma, J. G. Mesquita and P. N. Seminario-Huertas, Smooth dynamics of weakly damped Lamé systems with delay, SIAM J. Math. Anal. (to appear).