Conference: Mathematics of the Hemisphere

Opening

Chris Cosner, University of Miami: Reaction-diffusion-advection models with multiple movement modes

Classical reaction-diffusion-advection models for population dynamics with dispersal assume that all individuals move in the same way all the time. Actually, animals may switch between faster movement when searching for resources and slower movement while exploiting them, or juveniles may move differently than adults. To describe such situations requires systems of reaction-diffusion-advection equations. The resulting systems may have features different from single equations. In models based on logistic equations, the systems may be cooperative at low densities but competitive at high densities. It is well known that for a single diffusive logistic equation in a static spatially heterogeneous bounded domain, slower diffusion is advantageous. For stage structured populations where adults and juveniles have different environmental needs this is no longer always the case. This talk will describe some recent work on the theory and applications of reaction-diffusion-advection models for populations in bounded habitats where subpopulations may have different movement rates or patterns, and individuals can switch between subpopulations by behavior or contribute to them by reproduction and aging.

Shigui Ruan, University of Miami: Spatiotemporal Dynamics in Epidemic Models with Levy Flights: A Fractional Diffusion Approach

Recent field and experimental studies show that mobility patterns for humans exhibit scale-free nonlocal dynamics with heavy-tailed distributions characterized by Levy flights. To study the long-range geographical spread of infectious diseases, in this paper we propose a susceptible-infectious-susceptible epidemic model with Levy flights in which the dispersal of susceptible and infectious individuals follows a heavy-tailed jump distribution. Owing to the fractional diffusion described by a spectral fractional Neumann Laplacian, the nonlocal diffusion model can be used to address the spatiotemporal dynamics driven by the nonlocal dispersal. The primary focuses are on the existence and stability of disease-free and endemic equilibria and the impact of dispersal rate and fractional power on spatial profiles of these equilibria. A variational characterization of the basic reproduction number R₀ is obtained and its dependence on the dispersal rate and fractional power is also examined. Then R₀ is utilized to investigate the effects of spatial heterogeneity on the transmission dynamics. It is shown that R₀ serves as a threshold for determining the existence and nonexistence of an epidemic equilibrium as well as the stabilities of the disease-free and endemic equilibria. In particular, for low-risk regions, both the dispersal rate and fractional power play a critical role and are capable of altering the threshold value. Numerical simulations were performed to illustrate the theoretical results. (Based on G. Zhao & S. Ruan, J. Math Pures Appl. 2023).  Video

Ernesto Lupercio, CINVESTAV  Video

Xi Huo, University of Miami: Linking Mathematical Models to Mosquito Trap Data - A Study on Mosquito Population Dynamics in Miami-Dade

Aedes aegypti is a mosquito species responsible for a few arbovirus transmissions. In this talk, I will present how we connect differential equation parameters with the mosquito trap data collected from 2017 to 2019. We specifically will talk about parameter identifiability and model comparison problems we encountered during our fitting. The model is then used to compare the Ae. aegypti population and evaluate the impact of rainfall intensity in different urban built environments. Our results show that rainfall affects the breeding sites and the abundance of Ae. aegypti more significantly in tourist areas than in residential places. In addition, we apply the model to quantitatively assess the effectiveness of vector control strategies in Miami-Dade County in South Florida, USA. (Based on J. Chen, X. Huo, A. Wilke, J.C. Beier, C Vasquez, W. Petrie, R.S. Cantrell, C. Cosner, and S. Ruan, Acta Tropica 2023).  Video

Svetlana Rudenko, Florida International University: Solitary waves in KdV-type equations: stability vs. instability & blow-up

We discuss different nonlinear models, starting from Korteweg-de Vries equation (KdV), which describes shallow water waves in a canal, then consider its higher dimensional generalization Zakharov-Kuznetsov (ZK) equation, which was introduced in 1972 to describe 3d ion-acoustic waves in weakly magnetized plasma, then shift to Benjamin-Ono (BO) equation, that models 1d deep water waves, followed by  its 2d variant, the Shrira equation, describing the 2d boundary layer ocean waves. All these models are variants of the original KdV model, describing traveling solitary waves. We examine stability of solitary waves vs. instability and blow-up.  Video

Eduardo Teixeira, University of Central Florida: The Bernoulli problem with unbounded jumps

The Bernoulli problem appears naturally in mathematical models arising from fluid dynamics, cavitation, jet flows, optimal designs, to cite a view. The analysis of such models leads to a very rich class of free boundary problems. In this talk I will discuss Bernoulli free boundary problems prescribing unbounded jumps. The analysis is considerably more intricate as solutions are expected to be only Holder continuous, rather than Lipschitz as in the classical theory. The proof of the sharp regularity estimate and the analysis of fine geometric measure properties of the free boundary require several new ingredients. The ultimate goal is to classify the varying cups geometries along the free boundary.  Video

Ernesto Lupercio, CINVESTAV