Research

In ecology,  the spread of invasive species can be affected by interactions with other species, e.g., predators or biological control agents. Mosquito controls often rely on releasing large numbers of mosquitoes reared in the laboratory, which are either sterile or incapable of transmitting disease, to mate with wild mosquitoes and reduce their reproductive and vectorial capacity. The control succeeds if the indigenous population is replaced or eliminated over time. These correspond to certain steady states of the dynamical systems used to model the population dynamics. In my thesis, I discovered how the spatial distribution affects the performance of the control techniques by studying the model's asymptotic behaviors. 

Mathematical Results

• Mosquito population dynamics can be described in a continuous space with a reaction-diffusion system [1, 2, 5] or in a discrete patchy space with a metapopulation model [3].  The movement of the control agents is incorporated in the model through an inhomogeneous boundary condition [1] or a moving control function [2,5]. Dynamics of these spatial models pose interesting analytical challenges, which can be tackled by asymptotic methods, linear stability, and traveling wave analysis. 

• Mechanistic models can be combined with statistical methods for empirical data to identify biologically relevant parameter regimes [4].

The human brain is composed of many billions of interconnected neurons, which give rise to our mental lives and behaviors. The primary goal of system neuroscience is to understand how ensembles of neurons take in information, process it, and produce behaviors. This process is known as neural computation. A powerful framework to understand neural computation is to identify the neural dynamics - the rules that describe the temporal evolution of neural activity, often represented in a nonlinear dynamical system. Identifying neural dynamics from high-dimensional neural data is challenging and usually requires dimensionality reduction. Recently, the spectral properties of the Koopman operator have become promising for dimensionality reduction by the so-called Dynamic Mode Decomposition (DMD), due to its ability to expand the nonlinear flow as a linear combination of dominant coherent structures, and to extract spatial-temporal patterns from large-scale neural recording. 

Mathematical Results

• The first two dominant eigenfunctions of the Koopman operator associated with the limit cycle dynamics can be used to derive the phase-amplitude reduction of neural oscillation. Using this framework, we explain the interplay between the phase-resetting and the amplitude modulation of the neural activities in response to transient inputs [6].