December 18, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

April 12, 2013
9:00 a.m.- 5:00 p.m.
2013 Southern Ontario Dynamics Day

### SPEAKER ABSTRACTS

Cameron Browne, University of Ottawa
Pulse Vaccination in a Polio Meta-population Model

Poliomyelitis infections continue to arise in certain regions of Africa and the Middle East, despite the significant progress made toward the goal of eradication. There are several factors which contribute to the difficulty of eradication, including: areas of high transmission and low vaccination coverage, mobile populations, seasonality, and persistence of the virus in the environment. One of the main strategies utilized to fight Polio over the past 40 years has pulse vaccination.
In order to better understand this control strategy, we consider an impulsive $SIR$-type metapopulation model with seasonality, an environmental reservoir, and arbitrary pulse vaccination schedules in each patch. A basic reproduction number, $\mathcal R_0$, is defined and proved to be a global threshold for the system. Numerical calculations show the importance of, both, synchronizing the pulse vaccinations between the patches and the timing of the pulses with respect to the seasonality. A stochastic version of the model is also considered. It is found that pulse vaccination has a major advantage over a continuous vaccination strategy in terms of the probability of eradication. We also explore how the coupling between the patches and the level of environmental transmission affect the results.

Sue Ann Campbell, University of Waterloo
A Plankton Model with Delayed Nutrient Recycling

We consider a three compartment (nutrient-phytoplankton-zooplankton) model with nutrient recycling. When there is no time delay the model has a conservation law and may be reduced to an equivalent two dimensional model. We consider how the conservation law is affected by the presence of time delay (both discrete and distributed) in the nutrient recycling. We study the stability and bifurcations of equilibria when the total nutrient in the system is used as the bifurcation parameter. This is joint work with Matt Kloosterman and Francis Poulin.

Dong Eui Chang, University of Waterloo
Damping-Induced Self-Recovery Phenomenon In Mechanical Systems With An Unactuated Cyclic Variable

The falling cat problem has been very popular since Kane published a paper on this topic in 1969. A cat, after released upside down, executes a 180-degree reorientation, all the while having a zero angular momentum. It makes use of the conservation of angular momentum that is induced by rotational symmetry in the dynamics. But if there is an external force that breaks the symmetry, then the angular momentum will not be conserved any more.
Recently, we have discovered an exciting nonlinear phenomenon in mechanical systems with one unactuated cyclic variable on which a viscous damping force is exerted. In this case, there arises a new conserved quantity, called damping-added momentum, in place of the original momentum map. Using this new conserved quantity, we show that the trajectory of the cyclic variable asymptotically converges back to its initial value. This phenomenon, called damping-induced self-recovery, can occur even when the damping coefficient is not constant as long as the integral of the coefficient satisfies a certain condition.
The self-recovery phenomenon can be observed in a simple experiment with a rotating stool and a bicycle wheel which is a typical setup in physics classes to demonstrate the conservation of angular momentum. Sitting on the stool, one spins the wheel by his hand while holding it horizontally. A reaction torque will be created to initiate the rotational motion of the stool to the opposite direction. After some time, if the person applies a braking force halting the wheel spin, then the stool will asymptotically return to its original position, tracing back its past path regardless of the number of rotations the stool has made, provided that there is a viscous damping force on the rotation axis of the stool.

Hermann Eberl, University of Guelph
How we failed to solve an optimal control problem for a biofilm reactor with suspended growth

We present an extension of Freter's 1983 model of a continously stirred bioreactor with wall attachement (or equivalently: a biofilm reactor with suspended growth), accounting for substrate gradients in the microbial depositions. This is achieved by linking a CSTR model with a Wanner-Gujer like biofilm model. In the general multi-species, multi-substrate case this leads to a free boundary value problem for a coupled hyperbolic-parabolic problem which is not easily accessible to analytical techniques. However, in the simpler single species, single substrate case we can formally rewrite the model as a system of three ODEs, the longterm behavior of which can be studied with elementary techniques. In the second part of the talk we study the problem of controling the reactor such that a given amount of substrate is degraded as much as possible in as short a time as possible. This is joint work with Alma Masic (Lund University).

Xu Fei, Wilfrid Laurier University
Interspecific strategic effects of mobility in predator-prey systems

In this talk, we will investigate the dynamics of a predator-prey system with the assumption that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught. Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes.

Bernhard Lani-Wayda, Mathematisches Institut der Universit\"at Giessen
Four-dimensional \v{S}il'nikov-type dynamics in x'(t) = -\alpha \cdot x(t-d(x_t))

It was shown by Hans-Otto Walther that the state-dependent delay function $d$ in the linear-looking' equation $x'(t) = -\alpha \cdot x(t-d(x_t))$ can be chosen such that it produces a solution homoclinic to a saddle. Chosing proper subsets from the domain of a return map, and using the fixed point index, we prove that the return map is semi-conjugate to the shift in two symbols. The situation is an infinite-dimensional analogue to the one described by \v{S}il'nikov (1967) in dimension 4, and our methods are much inspired by the work of W\'{o}jczik and Zgliczy\'{n}ski on covering relations. Joint work in progress with Hans-Otto Walther.

Jean-Philippe Lessard, Université Laval
Rigorous Computations for Infinite Dimensional Problems

Studying and proving the existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for partial differential equations, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. As a consequence of these challenges and with the recent availability of powerful computers and sophisticated software, numerical simulations quickly became one of the primary tool used by scientists to conjecture the behaviour of the dynamics of the above mentioned nonlinear equations. A standard approach adopted by mathematicians is to get insights from numerical simulations to formulate new conjectures, and then attempt to prove the conjectures using pure mathematical techniques only. As one shall argue, this strong dichotomy need not exist in the context of dynamical systems, as the strength of numerical analysis and functional analysis can be combined to prove, in a direct computational way, existence of solutions of infinite dimensional dynamical systems. The goal of this talk is to present such rigorous numerical methods to the context of proving the existence of steady states, time periodic solutions, traveling waves and connecting orbits of finite and infinite dimensional differential equations.

F.M.G. Magpantay, York University
An Age-Structured Population Model with State-Dependent Delay

We considered an age-structured population model with distinct juvenile and adult stages in which the two stages consume different limited food sources. The new model involves the McKendrick PDE, a nonlinear boundary condition due to the birth rate, and a threshold condition with state-dependent delay due to a varying age of maturity. We present some analysis on this model and compare it to a version with constant delay as well as other existing population models. This is a joint work with N. Kosovali´c and J. Wu.

Daniel Munther, York University
The ideal free strategy with weak Allee effect

This talk examines the interplay between optimal movement strategies and the weak Allee effect within the context of two competing species in a spatially heterogenous environment. When both species have the same populations dynamics, previous studies identified an ideal free' strategy which is able to exclude any other competitor playing a
non-ideal free' strategy. We find that if the ideal free disperser is subject to a weak' Allee effect, a competing species utilizing very weak or very strong advection will still be excluded despite having superior population dynamics. However, for intermediate advection rates, such a competitor can invade the ideal free disperser and even drive it to extinction. Not only do these results enhance ecological understanding of competing species, but they provide insight into the non-linear theory of reaction-advection-diffusion models when the usual linearization techniques offer no information.

Felix Njap, University of Waterloo
Bifurcation analysis of a model of Parkinsonian STN-GPe activity

Excessive oscillations in the beta (15-30Hz) band are seen in the basal ganglia of patients with Parkinson’s disease. An important question concerns the conditions under which theseoscillations can occur. The mathematical technique of bifurcation analysis is applied to a simple model of 2 populations to determine the critical boundaries in parameter space. These boundaries separate regions of different dynamics. A number of bifurcations (up to codimension 2) are found. In particular, a region of beta oscillations has been identified in the model under “Parkinsonian conditions”. No such region is present under “healthy conditions”.

Longxing Qi, Anhui University and York University
Modeling The Schistosomiasis on the Islets in Nanjing

A compartmental model is established for schistosomiasis infection in Qianzhou and Zimuzhou, two islets in the center of Yangtzi River near Nanjing, P. R. China. The model consists of five differential equations about the susceptible and infected subpopulations of mammalian Rattus norvegicus and Oncomelania snails. We calculate the basic reproductive number R0 and discuss the global stability of the disease free equilibrium and the unique endemic equilibrium when it exists. The dynamics of the model can be characterized in terms of the basic reproductive number. The parameters in the model are estimated based on the data from the field study of the Nanjing Institute of Parasitic Diseases. Our analysis shows that in a natural isolated area where schistosomiasis is endemic, killing snails is more effective than killing Rattus norvegicus for the control of schistosomiasis.

Gunog Seo, York University
A comparison of two predator-prey models with Holling type I functional response

In my talk, I will show dynamics of two models, a laissez-faire predator-prey model and a Leslie-type predator-prey model, with type I functional response. For both models, I will study stability of the equilibrium where the prey and predator coexist by performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, in particular, I use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. I numerically show that my two models can possess two limit cycles surrounding a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit supercritical and discontinuous Hopf bifurcations. I then present and analyze a new functional response, built around the arctangent, that smooths the sharp corner in a type I functional response. For the new functional response, both models exhibit Hopf, cyclic-fold, and Bautin bifurcations.

Steve C. Walker, McMaster University
When does intraspecific variation matter to community-level dynamics?

Community ecologists typically ignore intraspecific variation, thereby implicitly assuming that each individual of a species is identical. This approach is reasonable, given the expenses associated with collecting enough data to estimate differences between conspecifics. Here we incorporate intraspecific variation into models of multispecies dynamics, in order to identify conditions under which ignoring intraspecific variation will lead to incorrect deductions about the dynamics of aggregate community properties (e.g. total biomass; community-level trait variance). Our approach is based on Price's equation from evolutionary biology and laws of total central moments from probability theory. This approach allows us to partition the dynamical equations for aggregate community properties into two additive components: one that ignores intraspecific variation and one that does not. We find that ignoring intraspecific variation can lead to dramatically incorrect deductions in many, but not all, circumstances.

Gail S. K. Wolkowicz, McMaster University
Mathematical model of anaerobic digestion in a chemostat:effects of syntrophy and inhibition

Anaerobic digestion is a complex naturally occurring process during which organic matter is broken down to biogas and various byproducts in an oxygen-free environment. It is used for waste and wastewater treatment and for production of biogas, especially methane. A system of differential equations modelling the interaction of microbial populations in a chemostat is used to describe three of the four main stages of anaerobic digestion: acidogenesis, acetogenesis, and methanogenesis. To examine the effects of the various interactions and inhibitions, we first study an inhibition-free model and obtain results for global stability using differential inequalities together with conservation laws. These results are compared with the predictions for the model with inhibition. In particular, inhibition introduces regions of bistability and stabilizes some equilibria.

Jianhong Wu, York University
Emerging flocking behaviors of the Cucker-Smale model with delayed information processing

This is based on the joint work with Yicheng Liu of the National University of Defense Technology. We examine the emerging behaviors such as flocking, herding and schooling in the Cucker-Smale model with delayed information processing in self-organized systems. We use a fixed point theoretic argument in weighted Banach spaces to derive sharp conditions on the influence function to ensure unconditioning flocking.

Yanyu Xiao. York University
Can treatment effect the epidemic final size?

Antiviral treatment is one of the key pharmaceutical interventions against many infectious diseases. This is particularly important in the absence of preventive measures such as vaccination. However, the evolution of drug-resistance in treated patients and its subsequent spread to the population pose significant impediments to the containment of disease epidemics using treatment. Previous models of population dynamics of influenza infection have shown that in the presence of resistance, the epidemic final size (i.e., the total number of infections throughout the epidemic) is affected by the treatment rate. These models, through simulation experiments, illustrate the existence of an optimal treatment rate, not necessarily the highest possible rate, for minimizing the epidemic final size. However, the conditions for the existence of such an optimal treatment rate have never been found. Here, we provide these conditions for a general class of models covering previous literature, and investigate the combination effect of treatment and transmissibility of the resistant strain on the epidemic final size. For the first time, we obtain the final size relations for an epidemic model with two strains of a disease (i.e., drug sensitive and drug resistance). We discuss the general model with specific functional forms of resistance emergence, and illustrate the theoretical findings with numerical simulations.

Hossien Zivari-Piran, York University
Numerical qualitative analysis of a large-scale model for measles spread

In this talk I describe the dynamic of a model of measles infection developed by Heffernan and Keeling (2008). The combined immunological and epidemiological model includes waning immunity and vaccination and is formulated as a large-scale system of ODEs. I show how the dynamics of this system changes as the parameters controlling waning time and vaccination level vary. The relevant local and global bifurcations are investigated using special numerical methods for large-scale systems, leading to a novel approach for explaining the mechanisms underlying the oscillatory dynamic of measles.

Xingfu Zou, University of Western Ontario
Can multiple Malaria species co-persist? ----Dynamics of multiple malaria species

There are several species of malaria protozoa spreading in different regions. On the other hand, the world becomes more highly connected by travel than ever before. This raises a natural concern of possible epidemics caused by multiple species of malaria parasites in one region. In this study, we use mathematical models to explore such a possibility. Firstly, we propose a model to govern the within-host dynamics of two species. Analysis of thismodel practically excludes the possibility of co-persistence (or super-infection) of the two species in one host. Then we move on to set up another model to describe the dynamics of disease transmission between human and mosquito populations without the co-infection class (using the results for the within-host model). By analyzing this model, we find that co-endemics of both species in a single region is possible within certain range of model parameters. This is a joint work with Yanyu Xiao.