Réunion d'hiver SMC 2024

Vancouver/Richmond, 29 novembre - 2 decembre 2024

Recent Advances in Differential Equations and Applications
Org: Elena Braverman (University of Calgary) et Kunquan Lan (Toronto Metropolitan University)
[PDF]

ELENA BRAVERMAN, University of Calgary On logistic models incorporating various diffusion strategies with and without harvesting [PDF]: Defining a diffusion strategy as the tendency to have a distribution proportional to a certain positive prescribed function, once a diffusion coefficient grows infinitely. We explore the interplay of harvesting and dispersal strategies and their influence on the outcome of the competition for two resourse-sharing species. While achieving extinction by excessive culling of the undesired species in many cases is a simple and efficient strategy, keeping biodiversity is a more complicated task. Proposing such heterogeneous harvesting that the two managed populations become an ideal free pair allows to guarantee coexistence. The directed movement is modeled by the term which particular form is $\Delta (u/P)$, where $P$ is the target distribution. However, when $P$ is not aligned with the carrying capacity of the environment, a unique positive solution $u^{\ast}$ of the Neumann problem is different from $P$. Another conclusion that we manage to deduce is that, once an invading species manages to mimic the observed distribution of the host species and has some advantage in the carrying capacity, this guarantees successful invasion. However, the conditions under which the host species can sustain, other than targeted culling of invaders or trimming both populations, is still an open question.
SUE ANN CAMPBELL, University of Waterloo Time Delays, Symmetry and Hopf Bifurcation in Oscillator Networks [PDF]: We consider networks of oscillator nodes with time delayed, global circulant coupling. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of phase-locked oscillations. We apply the theory to a variety of systems inspired by biological neural networks to show how Hopf bifurcations can determine the synchronization state of the network. Finally we show how interaction between two Hopf bifurcations corresponding to different oscillation patterns an induce complex torus solutions in the network.
YUMING CHEN, Wilfrid Laurier University An algebraic approach to determining negative (semi-)definiteness in applying the Lyapunov direct method [PDF]: In this talk, we first present a novel approach in determining the specific form of a Lyapunov function when the type of its candidate is given. Then we apply this approach to autonomous polynomial differential systems. Applications to some epidemic models described by polynomial differential systems indicate that the discussion on global stability can be greatly simplified.
QI DENG, York University Modeling the Interaction of Cytotoxic T-lymphocytes and Oncolytic Viruses in a Tumor Microenvironment [PDF]: Oncolytic virotherapy has become a promising approach in treating cancer. In this talk, we will discuss a mathematical model, which is developed to understand the interaction among immune cells, Oncolytic viruses, and tumor cells. The basic reproductive number ($R_0 $) is derived, and the local and global dynamics of the system are analyzed in terms of $R_0$ and another related threshold $R_0^E$. The theoretical results suggest that the system have periodic solutions and bifurcations. Numerical simulations further show that the immune response plays an obstructive role on virotherapy, and once the immune cell proliferation rate exceeds a threshold, the tumor will escape.
HERMANN EBERL, University of Guelph A spatio-temporal model of blossom blight [PDF]: Fireblight is a bacterial disease of apple and pear trees that can wipe out an entire orchard in one season. We present a model for fireblight during blooming season. It consists of two semi-linear PDEs that describe pathogen dispersal, which are coupled in each point of the domain to two ODEs that describe the host flowers. Numerical simulations suggest the existence of Travelling Waves which we then set out to prove using upper and lower solutions and a fix-point argument.
KUNQUAN LAN, Toronto Metropolitan University Have the classical Riemann-Liouville fractional integrals been fully understood before? [PDF]: In this presentation, I shall present the new notion of a generalized Riemann-Liouville (R-L) fractional integral and properties including its domain and range. The new notion and properties provide new insight and understanding into the classical R-L fractional integral and its properties. Based on the new generalized R-L fractional integral, when one intends to employ the semigroup property involving the classical R-L fractional integral operator, derivative of a second order fractional R-L fractional integral or a variety of first order fractional integral equations, one should use the generalized R-L fractional integral operator instead of using the classical R-L fractional integral operator. Therefore, some previous well-known results are not precise.
CHONGMING LI, Queen's University Evolutionary Stability of Bacterial Persister Cells [PDF]: We model the switching process of bacteria between antibiotic dormant features and normal active replication using an integro-reaction-diffusion-advection partial differential equation (PDEs). The PDE captures the impacts of epigenetic inheritance of metabolic state by implementing a non-local term that models a birth jump process. We prove the well-posedness of the non-local PDE model followed by the corresponding stability analysis of the positive steady-state solutions. Of primary interest is an extension of the model to a wider scenario of biological evolution by examining the evolutionarily stable strategies (ESSs) of persister cells. The idea is that genetic mutations will occasionally occur, and these mutations can alter any of the parameters describing the persister cell dynamics. As a first step we prove that, in a finite dimensional version of the model, the ESS strain is one that optimizes resource consumption irrespective of its pattern of dormancy. The next step will be to apply semigroup methods to the infinite dimensional system.
ZHISHENG SHUAI, University of Central Florida Impact of Incidence Functions on Epidemiological Model Dynamics: Mass Action vs. Standard Incidence [PDF]: The selection of incidence functions in epidemiological models plays a critical role in shaping the disease dynamics, particularly in populations of varying sizes. In this presentation, we examine a model that incorporates post-infection mortality and partial immunity, comparing the effects of mass-action and standard incidence functions. With the mass-action incidence, the model exhibits periodic solutions under certain parameter conditions. In contrast, applying the standard incidence reduces the likelihood of periodic solutions, potentially eliminating them entirely.
OLGA VASILYEVA, Memorial University of Newfoundland, Grenfell Campus Steady states and evolution of dispersal in river networks [PDF]: Steady states of nonlinear reaction-diffusion-advection (RDA) models can be viewed as solutions of a system of two first order ODEs (subject to appropriate boundary conditions). Geometrically, they are represented by orbits in the phase plane, generated by the corresponding flow operator. In this talk, I will discuss applications of the phase plane technique in a logistic RDA model in a river network setting. Here, a steady state is represented by a configuration of orbits in the corresponding phase plane satisfying geometric constraints induced by junction and boundary conditions. While in a single river case the basic shape of the steady state is determined by the boundary conditions, in the case of a river network, it is significantly affected by the geometry of the network (lengths of the segments and their cross-section areas). In a joint work with F. Lutscher and D. Smith, we exploit this phenomenon in the context of evolution of dispersal in a Y-shaped network. Namely, we consider the possibility of invasion of a steady state of a resident species by a species with different diffusivity. It turns out that the outcome of this interaction depends on the geometry of the network as well.
GAIL WOLKOWICZ, McMaster University Decay Consistent Models of Growth, Competition, and Predation [PDF]: Incorporating delay in various models of population interactions will be explored including competition and predation. In all of the models, any terms representing growth of a population take into consideration that individuals that do not survive, do not contribute to the growth of the population. The models are formulated so that survival is consistent with the decline terms in the model.
YUANXI YUE, Memorial university of Newfoundland Traveling wavefronts for the Belousov-Zhabotinsky system with non-local delayed interaction [PDF]: This talk presents a novel investigation into propagation dynamics of the Belousov-Zhabotinsky system with non-local delayed interaction, which exhibits dynamical transition structure from bistable to monostable. We address the enduring open problem on the existence, uniqueness and the speed sign of the bistable traveling waves. In the monostable case, we introduce new results for the minimal wave speed selection, which, as an application, further improved the existing investigations on pushed and pulled wavefronts. Our results can provide new estimate to the minimal speed as well as to the determinacy of the transition parameters. Moreover, these results can be directly applied to standard localized models and delayed reaction diffusion models by choosing appropriate kernel functions.
KEXUE ZHANG, Queen's University Input-to-State Stability in Terms of Two Measures [PDF]: Stability in terms of two measures encompasses a range of stability notions, such as partial stability, conditional stability, eventual stability, and practical stability. The concept of input-to-state stability (ISS) captures how external inputs affect the stability of control systems and has proven highly effective for stability analysis and control design in dynamical systems. This talk focuses on the two-measure version of ISS for nonlinear systems. We introduce various stability concepts for nonlinear systems and then explain the two-measure stability concept. This unified approach encourages us to examine ISS within the context of two measures, enabling analysis of external inputs' effects on entire states, partial states, periodic orbits, invariant sets, and more. Finally, we present sufficient conditions to ensure two-measure ISS for nonlinear systems and discuss potential applications of this approach.

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