Graduate Students Seminar

Speaker 1: Naghmeh Akhavan

Title

Multiple "die-out" Lyapunov functions for Lotka-Volterra models with population collapse

Abstract

We present a general trophic Lotka-Volterra model for which we show there is a bounded trapping region to which every trajectory is attracted. We explore a situation where this model satisfies a generalized competitive exclusion principle that applies so that one or more species will die out.  When a Lotka-Volterra competition model has no steady-state where all the species coexist, how many species must die out? Lyapunov functions can be vital tools for determining the behavior of a system. But constructing a Lyapunov function is usually difficult. Therefore, a general Lyapunov function for the d-dimensional competitive Lotka-Volterra model is still unknown. We find a novel type of Lyapunov function to show that "excess" species die out exponentially fast.

Speaker 2: Neha Agarwala

Title

Characterization and estimation of high dimensional sparse Gaussian regression parameters under closed polyhedral cone constraints

Abstract

We consider estimation of parameters in a high dimensional Gaussian regression setting where the parameters belong to a closed polyhedral cone.  The polyhedral cones are defined as the solution set of a homogeneous set of linear inequalities. A special case is when the number of restrictions maybe higher than the number of parameters. One such situation arises in estimation of monotone curve using a non-parametric approach e.g. splines.  In these high dimensional problems one usually seeks a `sparse' solution.  We define a notion of sparsity for such conic restrictions using lower dimensional facets of the cone. We also propose a sparse estimator of the constrained parameter vector by invoking a continuous shrinkage prior through the  higher dimensional non-negative orthant representation of the polyhedral cone.