Topics in Mathematical Biology: Theory, Applications and Future Perspectives
Org:
Kunquan Lan (TMU),
Gunog Seo (Colgate University) and
Gail S. K. Wolkowicz (McMaster)
[
PDF]
 STEPHANIE ABO, University of Waterloo
Can the clocks tick together despite the noise? Stochastic simulations and analysis [PDF]

The suprachiasmatic nucleus (SCN), also known as the circadian master clock, consists of a large population of oscillator neurons. Together, these neurons produce a coherent signal that drives the body’s circadian rhythms. What properties of the celltocell communication allow the synchronization of these neurons, despite a wide range of environmental challenges such as fluctuations in photoperiods? To answer that question, we present a meanfield description of globally coupled neurons modeled as Goodwin oscillators with standard Gaussian noise. Provided that the initial conditions of all neurons are independent and identically distributed, any finite number of neurons becomes independent and has the same probability distribution in the meanfield limit, a phenomenon called propagation of chaos. This probability distribution is a solution to a VlasovFokkerPlanck type equation, which can be obtained from the stochastic particle model. We study, using the macroscopic description, how the interaction between external noise and intercellular coupling affects the dynamics of the collective rhythm, and we provide a numerical description of the bifurcations resulting from the noiseinduced transitions. Our numerical simulations show a noiseinduced rhythm generation at low noise intensities, while the SCN clock is arrhythmic in the high noise setting. Notably, coupling induces resonancelike behavior at low noise intensities, and varying coupling strength can cause period locking and variance dissipation even in the presence of noise.
 JULIEN ARINO, University of Manitoba
Backward bifurcation in an SLIARS model with vaccination [PDF]

Backward bifurcations are known to occur in a variety of epidemic models when treatment or vaccination are taken into account. In a backward bifurcation situation, there can exist subthreshold endemic equilibria, leading to a bistable situation where the behaviour of the system is initial conditionsdependent. The situation is easy to characterise in a deterministic setting, but more complicated in a stochastic one. To investigate this, we considered an endemic (i.e., including demography) SLIARS model for a disease presenting symptomatic and asymptomatic stages, to which we added vaccination. We showed that the deterministic version of the model can undergo backward bifurcations. We then studied numerically the equivalent (stochastic) continuoustime Markov chain and observed that a backward bifurcation was observable also in this context. Interestingly, a branching process approximation of the stochastic process was unable to pick up the same characteristic. This is joint work with Evan Milliken (U of Louisville).
 MARYAM BASIRI, University of Ottawa
Traveling wave solutions for a free boundary problem modeling spread of ecosystem engineers [PDF]

Ecosystem engineers are species that modify their environment to make it (more) suitable for
them. Beavers are a wellknown engineering species. We present a novel model for the spread
of ecosystem engineers as a free boundary problem: ahead of the front, the habitat is unsuitable
for the species, and behind the front, the habitat is suitable. The engineering action of the
population moves the boundary ahead. We derive a semilinear parabolic equation from an
individual random walk model. The condition for the moving boundary is a biologically derived
twosided condition that models the movement behavior of individuals at the boundary as well as
the process by which the population moves the boundary to expand their territory.
We study the model with the Allee growth function and prove the existence of traveling wave
solutions of different types. Then we will determine how model parameters affect the ability of
an ecosystem engineer species to invade new environments and the speed at which such an
invasion occurs.
 ELENA BRAVERMAN, University of Calgary
On stability and asymptotics of equations and systems of population dynamics with concentrated and distributed delays [PDF]

Many differential equations of mathematical biology assume delayed production process and the instantaneous mortality. It is well known that introduction of delay can destroy stability of the unique positive equilibrium and even lead to chaos. However, for some types of equations and systems, lags in the reproduction term do not change stability properties. Consideration of variable, unbounded and distributed delays emphasizes robustness of this `absolute stability' property.
Some interesting phenomena are observed in equations of population dynamics when the production part includes two different delays. We also consider Hutchinson and MackeyGlass equations a controlled with a term whose variable coefficient can change its sign.
 ROBERT STEPHEN CANTRELL, University of Miami
A twostage reactiondiffusion system [PDF]

It is well known that in reactiondiffusion models for a single unstructured population in a bounded, static, heterogeneous environment, slower diffusion is advantageous. That is not necessarily the case for stage structured populations. In earlier work, Cantrell, Cosner and Martinez showed that in a stage structured model introduced by Brown and Lin, there can be situations where faster diffusion is advantageous. In this paper we extend and refine these results on persistence to more general combinations of diffusion rates and to cases where either adults or juveniles do not move. We also obtain results on the asymptotic behavior of solution as diffusion rates go to zero, and on competition between species that differ in their diffusion rates but are otherwise ecologically identical. We find that when the spatial distributions of favorable habitats for adults and juveniles are similar, slow diffusion is still generally advantageous, but if those distributions are different that may no longer be the case. This talk is based on joint work with Chris Cosner and Rachidi Salako.
 YUMING CHEN, Wilfrid Laurier University
Threshold dynamics of a viral infection model with defectively infected cells [PDF]

In this talk, we consider a viral infection model with defectively infected cells. We show that the basic reproduction number serves as a threshold parameter. When the basic reproduction number is less than or equal to unity, the infectionfree equilibrium is globally asymptotically stable; when it is larger than unity, the infection equilibrium is globally asymptotically stable. The stability is established by Lyapunov's direct method. Here we provide a procedure to determine whether the derivative of a given type of Lyapunov function candidate is negative (semi)definite or not.
 TIANYU CHENG, University of Western Ontario
A new perspective on infection forces with demonstration by a DDE infectious disease model [PDF]

In this work, we revisit the notion of infection force from a new angle which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain
many existing infection force functions in the literature, it can also motivate new forms of infection force functions, particularly infection forces depending on disease surveillance of the past.
As a demonstration, we propose a SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, particularly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of the past epidemics in the infection force. We confirm Hopf bifurcations both theoretically and numerically. The results show that, depending on how recent the disease surveillance data are, their assigned weight
may have a different impact on disease control measures.
 MONICA COJOCARU, University of Guelph
Individual risk and discomfort perceptions, NPI policies and the evolution of the pandemic in Ontario 2020 [PDF]

In this work, we provide a granular view of factors affecting COVID19 disease transmission across Ontario, Canada and the 34 public health units composing it. We estimate the perceived risk of infection and perceived personal discomfort of complying with nonpharmaceutical interventions (NPIs) in each PH region. With the use of dynamic programming and a simple NAsh game model, we estimate the expected NPI adoption proportion across Ontario from March to December 2020. Finally, we use an SEIRL compartmental model for Ontario to study the interplay between the estimated NPI adoption from the game and the actual evolution of the infection. Finally, we explore the limitations of our work, discuss the success of our computations and highlight possible avenues of further refinement.
This is joint work with: Sarah Smook (U. Guelph), David Lyver (U. Guelph) and Edward W. Thommes (U. Guelph)
 TROY DAY, Queen's University
The Epidemiology and Economics of Physical Distancing during Infectious Disease Outbreaks [PDF]

People's incentives during an infectious disease outbreak influence their behaviour, and this behaviour can impact how the outbreak unfolds. Early on during an outbreak, people are at little personal risk of infection and hence may be unwilling to change their lifestyle to slow the spread of disease. As the number of cases grows people may then voluntarily take extreme measures to limit their exposure. Political leaders also respond to the welfare and changing desires of their constituents. In this talk I will use ideas from the study of differential games to model how individuals’ and politicians’ incentives change during an outbreak. Motivated by the COVID19 pandemic, I focus on physical distancing behaviour and the imposition of stayathome orders. I show that the dynamic game being played in the population and its consequences are very different depending on the degree of asymptomatic transmission.
This is joint work with David McAdams, Fuqua School of Business and Economics Department, Duke University.
 HERMANN EBERL, University of Guelph
Chaos in the Hive and Beyond: A Multiscale Model of Nosemosis in an Apiary [PDF]

Recent years have seen the emergence of many generic or specific models for the dyamics of honeybee diseases. The vast majority of these studies consider a single hive only. We present and discuss an ecoepidemiological multiscale model of the transmission of Noesemosis in an apiary. For the transmission of the disease on the apiary level, i.e between hives, we develop a mathematical model of drifting, i.e. ``{\it the change of residence of bees from one hive to another}'' (as Corkins put it in 1932). For the transmission of the disease within a colony, we consider two routes, a direct and an indirect one. This extends our previous work on Nosemosis to the metapopulation setting. It leads to a system of $5N$ nonautonouos ODEs, where $N$ is the number of colonies considered. We explore the model numerically. The main finding is that the apiary level dynamics of the disease can be greatly different (highly irregular chaos vs periodic), depending on which of the two withinhive transmission routes dominates. This is joint work with Nasim Muhammad.
 SAMUEL MATTHIAS FISCHER, Osnabrück University & Helmholtz Centre for Environmental Research – UFZ
KDElikelihood: a tool for fitting stochastic dynamic models to equilibrium data [PDF]

Stochastic dynamic models are a valuable tool to study the spatial distribution of species and estimate their responses to disturbances and environmental changes. However, fitting such models to observational data can be challenging, because their complexity typically hinders direct application of classical statistical tools such as the likelihood. Hence, modellers often examine the parameter space by applying samplingbased methods (e.g. approximate Bayesian computation, ABC), or they consider aggregated results whose distribution may be approximated via the central limit theorem. However, if the considered species distributions are assumed to be in equilibrium state, reaching these states in simulations requires long runs, making methods such as ABC difficult to apply. At the same time, aggregating results may lead to information loss that could result in identifiability issues corrupting the reliability of the parameter estimates. In this talk, we suggest the kerneldensityestimatebased (KDE) likelihood as a tool circumventing these issues. The KDElikelihood allows modellers to exploit the favourable statistical properties of the likelihood function without deriving it in closed form. We showcase the method's advantages in real applications by modelling the distribution of trees in a temperate Chinese forest using the processbased model Formind.
 FRÉDÉRIC HAMELIN, Institut Agro
The proportion of resistant hosts in mixtures should be biased towards the resistance with the lowest breaking cost [PDF]

Current agricultural practices facilitate emergence and spread of plant
diseases through the wide use of monocultures. Host mixtures are a promising
alternative for sustainable plant disease control. Their effectiveness can be partly
explained by priminginduced crossprotection among plants. Priming occurs when
plants are challenged with noninfective pathogen genotypes, resulting in increased
resistance to subsequent infections by infective pathogen genotypes. We
developed an epidemiological model to explore how mixing two distinct resistant
varieties can reduce disease prevalence. We considered a pathogen population
composed of three genotypes infecting either one or both varieties. We found that
host mixtures should not contain an equal proportion of resistant plants, but a
biased ratio (80 : 20) to minimize disease prevalence. Counterintuitively, the
optimal ratio of resistant varieties should be biased against the costliest resistance
for the pathogen to break. This benefit is amplified by priming. This strategy also
prevents the invasion of pathogens breaking all resistances.
This is joint work with Pauline Clin, Frédéric Grognard, Didier Andrivon, and Ludovic Mailleret.
 CHRISTOPHER M HEGGERUD, UC Davis
Mathematical comparison and empirical review of the Monod and Droop forms for resourcebased population dynamics [PDF]

Almost all biological models use either the Droop or Monod form to describe the resourcebased growth
of a living organism. Empirical evidence overwhelmingly suggests the Droop form describes data more
accurately than the Monod form, however, the Monod form is more popular due to its simplicity. Focusing
on phytoplankton, we illustrate the underlying logics behind these two forms via conceptual comparison,
experimental data validation, transient, and asymptotic dynamics. The conceptual illustration provides the
primary difference in their mechanisms via a paradox. Data validation is tested via field and laboratory
experiments. The Droop and Monod forms have consistent asymptotic dynamics in the closed nutrient case,
whereas the transient dynamics are significantly different when the nutrient uptake rate is small.
 YU JIN, University of NebraskaLincoln
Population dynamics in a habitat with a protection zone [PDF]

Protecting native species or endangered species has been an important issue in ecology. We derive a reactiondiffusion model for a population in a onedimensional bounded habitat, where the population is subjected to a strong Allee effect in its natural domain but obeys a logistic growth in a
protection zone. We establish threshold conditions for population persistence and extinction via the principal eigenvalue of an associated eigenvalue problem. We then obtain the influences of the protection zone on the longterm population dynamics under different boundary conditions and propose strategies for designing the optimal location of the protection zone in order for the population to persist in the long run.
 LAURENCE KETCHEMEN, University of Ottawa
Populations dynamics in fragmented landscapes under monostable and bistable growth dynamics [PDF]

Many biological populations reside in increasingly fragmented landscapes, where habitat quality may change abruptly in space. A reactiondiffusion model for a single species which grows and disperses in a onedimensional heterogeneous landscape is presented. The landscape is composed of two homogeneous adjacent patches with different diffusivities and net growth functions (monostable and bistable). An interface condition connects population density and flux between the two patches. We first classify all possible positive steady states using a phase plane approach. We continue by analyzing the stability properties of certain simple possible positive steady states. We end by applying bifurcations theory. Numerical simulations reveal fold bifurcations.
 RONGSONG LIU, University of Wyoming
An Agestructured Model of Bird Migration [PDF]

An approach to modelling bird migration is proposed, in which there is a region where birds do not move but spend time breeding. Birds leave this breeding region and enter a migration flyway which is effectively a oneway corridor starting and ending at the breeding location. Mathematically, the flyway is a curve parametrised by arclength. Flight speed depends on position along the flyway, to take account of factors such as wind and the pausing of birds at various locations for wintering or stopovers. Percapita mortality along the flyway is both position and agedependent, allowing for increased risks at stopover locations due to predation, and increased risks to immature birds. A reactionadvection agestructured equation models population dynamics along the flyway and, using a Laplace transform technique, the model is reduced to a scalar delay differential equation for the number of adult birds at the breeding location. Extinction and persistence criteria are obtained for the bird population and the results of computer simulations are presented.
 XINZHI LIU, University of Waterloo
Multigroup flocking control of multiagent systems [PDF]

As a fascinating collective behavior, flocking is observed in a variety of biological species such as beasts, birds, fish, bees, and ants. In such situations, each individual acts as an autonomous agent and interacts only with its nearby neighbors, while the entire group displays coordinated behavior and can accomplish very complex tasks. Inspired by such collective intelligence of animal groups in nature, there has been an increased research interest in flocking control of multiagent systems around the world in recent years. This talk discusses the flocking control problem of multiagent systems with multigroup tracking various virtual leaders. Hybrid protocols are proposed to take into consideration of continuous communications among agents and intermittent information exchanges at a sequence of discrete times. It is shown, by employing results from graph theory and dynamical systems, that agents may be divided into multiple subgroups to follow different leaders while maintaining desired subformation configurations as well as collisionfree motions.
 CHUNHUA OU, Memorial University of Newfoundland
Dynamics of Diffusive LotkaVolterra competition systems in a shifting environment [PDF]

We studied the existence, uniqueness and stability of forced traveling waves for the LotkaVolterra competition system in a shifting habitat. Based on the asymptotic behaviors of the wave profiles and by means of upperlower solution method coupled with sliding technique, we showed that the forced wave for the system exists and is unique, when the forced speed lies in a specific interval. Explicit expressions of the two end points of this interval were derived and our finding indicated that they are related to the FisherKPPtype invasion speed. Furthermore, we established a squeezing theorem to show the local stability of the forced waves. With the aid of comparison principle and Xinfu Chen's idea (1997), we established the global stability of the forced waves when the initial data were properly assigned. Finally, a gap formation between two species were studied when forced traveling wave doesn't exist.
 WEIWEI QI, University of Alberta
Noiseinduced transient dynamics [PDF]

Many complex processes exhibit transient dynamics  intriguing or important dynamical behaviors over a relatively long but finite time period. A fundamental issue is to understand transient dynamics of different mechanisms. In this talk, we focus on a class of randomly perturbed processes arising in population dynamics where species only persist over finite time periods and go to extinction in the long run. To capture such transient persistent dynamics, we use quasistationary distributions (QSDs) and study their noisevanishing asymptotic. Special attention will be paid to essential differences between models with and without environmental noises. The talk ends up with some discussions.
 SHIGUI RUAN, University of Miami
Imperfect and BogdanovTakens Bifurcations in Biological Models: From Harvesting of Species to Removal of Infectives [PDF]

The bifurcation induced by small perturbations of a system (on already existed bifurcations) that leads to more complex bifurcations simplifying into separate saddle and node equilibrium points is called an imperfect (perturbed) bifurcation. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predatorprey models with stocking/harvesting and epidemic models with immigration/isolation. First we consider a predatorprey model with Holling type II functional response whose dynamics and bifurcation are wellunderstood. We will show that that introducing constant stocking/harvesting of predators induces imperfect bifurcation: For the case with stocking, the model has one positive equilibrium and one negative equilibrium when stocking constant increases from zero. For the case of harvesting, the model has none, one, or two positive equilibria when harvesting constant varies; then we explain that the unique positive equilibrium is a cusp of codimension 2 and the model undergoes BogdanovTakens bifurcation. We also consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and BogdanovTakens bifurcations when the constant perturbation rate varies. (Based on a joint work with Dongmei Xiao).
 CHUNHUA SHAN, The University of Toledo
Transmission dynamics and periodic phenomena in a model of West Nile virus with maturation time [PDF]

West Nile virus is a typical vectorborne disease transmitted to humans and animals by Culex mosquitoes, where avian birds serve as amplification hosts for the virus. To investigate the role of mosquitoes in the transmission dynamics of West Nile virus, we formulate a system of delay differential equations with a standard incidence rate to model the interaction between mosquitoes and birds. We show that the maturation time of mosquitoes affects disease transmission in sophisticated ways. It turns out that a large maturation delay will lead to the extinction of mosquitoes and the disease, a small maturation delay will stabilize the epidemic level of the disease, and an intermediate maturation delay will cause sustainable oscillations of mosquito population, recurrence of diseases, and even mixedmode oscillation with an alternation between oscillations of large and small amplitudes.
 ZHONGWEI SHEN, University of Alberta
Coexistence in random environments [PDF]

We study the evolution of a community modelled by systems of stochastic differential equations with demographic noises, which exhibit coexistence dynamics only over finite periods. To capture such coexistence dynamics, we study the solution process before one species goes extinct and use quasistationary distributions (QSDs) to capture dynamical states governing the coexistence. In this talk, I will report some ideas and recent progress about dynamical properties of QSDs.
 JUNPING SHI, College of William & Mary
Turing type bifurcation in reactiondiffusion models with nonlocal dispersal [PDF]

In reactiondiffusion models describing biological and chemical interactions, some dispersal and interaction can be of nonlocal nature. We show that when a nonlocal dispersal occurs instead of classical diffusion, how the mechanism of Turing diffusioninduced instability and pattern formation changes. It is shown that Turing type instability and associated spatial patterns can be induced by fast nonlocal inhibitor dispersal and slow activator diffusion, and slow nonlocal activator dispersal also causes instability but may not produce stable spatial patterns. The existence of nonconstant positive steady states is shown through bifurcation theory. This suggests a new mechanism for spatial pattern formation, which has a different instability parameter regime compared to the Turing mechanism. This talk is based on joint work with Shanshan Chen, Xiaoli Wang and Guohong Zhang.
 SABRINA H. STREIPERT, University of Pittsburgh
Introduction and Application of the Augmented Phase Portrait [PDF]

We introduce the nextiterate operators and corresponding nextiterate rootsets and rootcurves associated with the nullclines of discrete planar systems. The signs of these nextiterate operators are used to augment the standard phase portrait that includes the direction field and the nullclines, to determine whether a point is mapped above or below the corresponding nullcline. This method identifies positively invariant regions and regions that can give rise to periodic solutions. The construction of the augmented phase portrait is demonstrated using an example of a rational planar map that arises in population modeling. We show that the augmented phase portrait can provide an elementary, alternative approach for determining the global dynamics of this model. The potential and limitations of the augmented phase portrait are explored using several examples that have applications in population dynamics, epidemiology, and delay difference equations.
 HAO WANG, University of Alberta
Cognitive Animal Movement Modelling [PDF]

Based on our recent efforts in spatial memory modelling and analysis for cognitive animal movement, I will briefly review this topic and mention challenges in modelling and rigorous analysis. Some showcases will be given as idea illustrations.
 XIAOQIANG ZHAO, Memorial University of Newfoundland
Spatial Dynamics of Species with Annually Synchronized Emergence of Adults [PDF]

In this talk, I will report our recent research on the spatial dynamics of species growth with annually synchronous emergence of adults by formulating an impulsive reactiondiffusion model. With the aid of the discretetime semiflow generated by the oneyear solution map, we establish the existence of the spreading speed and traveling waves for the model on an unbounded spatial domain. It turns out that the spreading speed coincides with the minimal speed of traveling waves, regardless of the monotonicity of the birth rate function. We also investigate the model on a bounded domain with a lethal exterior to determine the critical domain size to reserve species persistence. Numerical simulations are illustrated to confirm the analytical results and to explore the effects of the emergence maturation delay on the spatial dynamics of the population distribution. In particular, the relationship between the spreading speed and the emergence maturation delay is found to be counterintuitively variable.
 HUAIPING ZHU, York University
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