My recent research mainly focuses on applications of partial differential equations to mathematical ecology, population genetics, and disease dynamics.
I have been interested in understanding the effects of dispersal on population dynamics via reaction-advection-diffusion models. We showed that the geometry of a habitat can play an important role in the evolution of conditional dispersal, and that strong directed movement of species can induce the coexistence of competing species. Most recently I have been studying the evolution of density-dependent dispersal. I have also been working on the optimal spatial arrangement of resources and how advection along resource gradients affects the extinction of species.
Part of my research is on the evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and selection, and we studied three models: (i)discrete space and discrete time; (ii)discrete space and continuous time; and (iii)continuous space and continuous time. These models yield, respectively, a system of (i)nonlinear, first-order difference equations; (ii)nonlinear, first-order differential equations; and (iii)semilinear, parabolic partial differential equations. Among the questions discussed are the loss of a specified allele, the maintenance of a specified allele or every allele, the existence and stability of completely polymorphic equilibria, the weak- and strong-migration limits, and uniform (i.e., location-independent) selection.
For disease dynamics, I am interested in spatial spread of diseases in heterogeneous environment, e.g., on the effects of spatial heterogeneity, habitat connectivity, and rates of movement on the persistence and extinction of infectious diseases. We have studied frequency-dependent SIS epidemic model with n patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates. We also studied a spatial SIS reaction-diffusion model, with the focus on the existence, uniqueness and profile of endemic equilibrium.