Minisymposium on “Differential Equations and Math Biology”
演讲时间：9:00am-12:00pm, August 10, 2015
演讲人：Xiaoqiang Zhao, Memorial University of Newfoundland, Canada
演讲题目（Title）: Spatial Invasion Threshold of Lyme Disease
报告摘要（Abstract）: A nonlocal model of Lyme disease is formulated to incorporate a spatially heterogeneous structure. The basic reproduction number R0 of the disease and its computational formula are established. It is shown that R0 serves as a threshold value between extinction and persistence in the evolution of Lyme disease. Numerical simulations indicate that spatial heterogeneity of the disease transmission coefficient increases the basic reproduction number, but spatial heterogeneity of the carrying capacity of mice alleviates the value of R0. Moreover, the influence of host population in size, destruction of tick habitats and deployment of vaccinations is studied to give insights into optimal control of the disease.
演讲人：Vlastimil Krivan, Faculty of Science USB, Czech Republic
演讲题目（Title）：On some ecological applications of differential equations with discontinuous righthand sides
报告摘要（Abstract）: In my talk I will show how models described by differential equations with dis continuous righthand sides arise in ecology. I will start with the work of Vito Volterra, who derived what is now known as the Lotka–Volterra predator-prey model (Volterra, 1926). Soon after its publication, this model was experimentally tested in a series of population experiments by G . F. Gause (Gause, 1934; Gause et al., 1936). The results were inconsistent with the Lotka–Volterra neutrally stable limit cycles. In one of his experiments with protists feeding on yeast there was experimental evidence that population dynamics tended to a limit cycle. These observations lead Gause et al. (1936) to search for discrepancies in assumptions of the Lotka–Volterra predator-prey model when applied to their experimental systems. They observed that protists were not able to feed on yeast at low densities, because at low yeast densities the prey formed into a sediment at the bottom which was not accessible to predators inhabiting the water column. When prey reached above the critical density, they re-appeared in the water column and became immediately accessible to predators. To describe their observations mathematically, Gause et al. (1936) substituted the linear consumption rate used in the Lotka–Volterra model by a saturating discontinuous function. This discontinuity leads to a predator-prey dynamics that do not have solutions in the classical sense. It is quite remarkable that using a geometrical argument, Gause et al. (1936) were able to predict that “trajectories” of their model converged to a limit cycle. A concept of a “solution” for such models was introduced later by A. F. Filippov (1988). No such mathematical concept existed at the time when Gause with his co-workers analyzed their model to achieve a better fit with their experimental data. In my talk I will re-analyze Gause et al. mode l (Kˇrivan, 2011; Kˇrivan and Pryiadarshi, 2015) and will show, how models with discontinuous righthand sides arise when evolutionary principles are applied to population dynamical mode ls of interacting populations. In particular, such differential equations often arise in mo dels that combine adaptive animal behaviors with population dynamics. If behavior changes on a fast time scale (relative to demography time scale) it can be described by the Nash equilibria (or ESS if they exist) of the underlying evolutionary game at the current population densities. This leads to a complex feedback between population abundance and behavior which, in turns, influences population dynamics. This feedback is often described by a differential inclusion (or by the Fillipov regularization of a differential equation with discontinuous righthand side).
Filipp ov, A. F., 1988. Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht. Gause, G. F., 1934. The struggle for existence. Williams and Wilkins, Baltimore. Gause, G. F., Smaragdova, N. P., Witt, A. A., 1936. Further studies of interaction between predators and prey. The Journal of Animal Ecology 5, 1–18. Kˇrivan, V., 2011. On the gause predator-prey model with a refuge: A fresh look at the history. Journal of Theore tic al Biology 274, 67–73. Krivan, V., Pryiadars hi, A. 2015. L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge. Journal of theoretical biology 370:21-26 Volterra, V., 1926. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560.
演讲人：Ming Mei, Champlain College St-Lambert & McGill University, Canada
演讲题目（Title）: Asymptotic behavior of solutions to reaction-diffusion equations with time-delay
报告摘要（Abstract）: In this talk, we consider a mono-stable reaction-diffusion equation with time-delay, which represents the population model of single species like Australian blowflies. When the system of equations is non-monotone, it possesses some monotone or non-monotone traveling waves dependent on the time-delay to be small or big. We clarify that, for a certain given initial data, the corresponding solution will converge to a certain monotone or non-monotone traveling wave, where the wave speed can be specified due to how fast the initial data vanishes at far field $x=-\infty$, the location of the wave can be also determined by the given initial data, and the shape of the wave is determined by the size of the time-delay.
演讲人：Wenrui Hao, Mathematical Sciences Institute, Ohio State University
演讲题目（Title）: Parameter investigation and biological systems
报告摘要（Abstract）: This talk will cover some recent progress on parameter study on nonlinear partial differential equations (PDEs) arising from biology. This parameter study is based on homotopy continuation method, and makes use of polynomial systems (with thousands of variables) arising by discretization. It includes bifurcation, parameter region for identifying the number of stable steady states, and risk map for treatment of biological diseases.