Research Projects

Phytoplankton Project

Postdoctoral project, Supervised by Professor Simon Levin, 2017

I am formulating a model for the dynamics of phytoplankton in a stratified lake. Stratification separates the water with a horizontal plane called thermocline into two zones: epilimnion and hypolimnion. The epilimnion is the upper zone which is warm (lighter) and well mixed. The hypolimnion is the bottom colder zone. It is usually dark, and relatively undisturbed. Most deep lakes on Earth are stratified. Phytoplankton, also called algae, grow via photosynthesis in use of nutrients such as phosphorus and nitrogen from water and energy from sunlight. Because the hypolimnion is not well mixed, the change in the phytoplankton density and nutrient concentration in it depends on time and depth in the water column. Phytoplankton can be moved from their position by tur- bulent mixing (diffusion) or by sinking (advection). The change in the phytoplankton density in the epilimnion is independent of the depth (since it is well mixed over night). Light absorption in a lake follows Lambert-Beer’s Law. This novel model is the hybrid of highly interconnected nonlinear partial and ordinary differential equations. I will later extend the model to a bacteria-phytoplankton interaction model in a stratified lake.

Biodegradation Project

Part of my PhD projects., University of Alberta, 2017

I have formulated and investigated a stoichiometric organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. I performed complete local and global stability analyses of the system and determined the criteria for the uniform persistence and extinction of the species and chemicals. Moreover, I determined the optimal value of grazers that maximizes degradation of organic matter. Based on the parameters under the control of the ex- perimenter, I performed one- and two-parameter bifurcation analyses. Furthermore, I determined the switching time of bacterial growth rate from carbon dependent to nitrogen dependent and vice versa for different continuous cultures. I determined the sensitivity of the degradation rate with respect to the model parameters. I also discussed numerically how the dilution rate affects the decomposition percentage as well as decomposition speed. In collaboration with M. Lewis (math & biological sciences), T. Siddique (renewable resources), J. Foght (biological sciences), Hao Wang (math) and industrial partners, we have modified this model to a greenhouse gas (GHG) generation model for oil sands settling basins and end pit lakes. First, I fitted the model to naphtha hydrocarbons in tailings from Syncrude’s Mildred lake settling basin to determine optimal parameter values. Next, I extended it to a GHG generation model and then validated the model using measured methane data from Syncrude’s mature fine tailings (MFT). This model is cross validated using the hydrocarbons and methane data from Shell Albian’s MFT.


Part of my PhD projects., University of Alberta, 2017

I have done substantial research on indirectly and directly transmitted infectious diseases. I formu- lated and analyzed a cholera transmission model, which explicitly includes the dynamics of bacte- riophage and bacteria, and also contains an indirect infection term which accounts for a minimum infectious dose of the pathogen V. cholerae. At the moment, I am working on a multi-scale model for Guppy-Gyrodactylus interactions. This is a collaborative research venture involving G. Fussmann’s lab (biology at McGill university) and my supervisor. In this first phase of the project we are formulating a guppy-gyrodactylus interaction model with distributed delay. The model consists of five coupled nonlinear differential equations tracing the rates of change of guppy population, guppy immune response and gyrodactylus population. The guppy total population is divided into three sub-groups: susceptible, infected, and recovered guppies. This project at the interface of math- ematics and experimental epidemiology attempts to provide an experimental test bed for spatial disease and host-parasite dynamics. The proposed research project provides a unique opportunity to establish an ethically acceptable and cost-effective explorative tool for general infectious disease dynamics. In the next phase of the project, we will modify it to a mathematical model that contains two nested meta-population levels. At the basal level, guppies serve as patches for parasites, which disperse from fish to fish; the higher level consists of fish groups in different patches that are linked through dispersal of parasitized or non-parasitized guppies

Bayesian Nonparametric Tensor Completion

Data Science, University of Alberta, 2017

In this project, we propose a Bayesian nonparametric method to estimate missing data in tensors. The proposed method uses a Tucker-1 factorization to learn a smaller core tensor and a factor matrix via Gibbs sampling. Unlike most existing Tucker factorization algorithms, the tensor rank is estimated automatically. This is done by employing an Indian Buffet Process prior over the latent matrix, where the estimated number of non-zero columns corresponds to the tensor rank. The missing data is then estimated based on the latent structure learned from the factorization. As a result, the method provides a form of Bayesian model averaging over estimated ranks, allowing it to avoid overfitting. As an illustration, a num- ber of datasets are artificially corrupted at various levels. Relative error is used to assess completion performance and our result is compared to parallel matrix factorization as a baseline. The empirical results show that our method does not beat the current state of the art, but provides pertinent information that other methods do not. That is, statistical quantities, like mean standard error, are readily available due to the Bayesian estimation framework of our method. Lastly, we identify some future avenues for research that may improve performance.