Ph.D. Public Lecture

Speaker: Kumar Shukla (Western)

"Complexity 0 Torus Actions on Manifolds"

Time: 10:00

Room: MC 107

Let T be an n-dimensional torus acting effectively on a 'nice' 2n-dimensional manifold M, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. We begin by collecting some general facts about actions of compact Lie groups on manifolds. Then we briefly discuss repre- sentation theory of tori and prove some facts about orbits and fixed-point sets of torus actions. Finally, using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that the quotient space M/T is a homology disk.

Ph.D. Public Lecture

Speaker: Yanni Zeng (Western)

"Population Dynamics and Bifurcations in Predator-Prey Systems with Allee Effect"

Time: 13:00

Room: zoom

This thesis investigates a series of nonlinear predator-prey systems incorporating the Allee effect using differential equations. The main goal is to determine how the Allee effect affects population dynamics. The stability and bifurcations of the systems are studied with a hierarchical parametric analysis, providing insights into the behavioral changes of the population within the systems. In particular, we focus on the study of the number and distribution of limit cycles (oscillating solutions) and the existence of multiple stable states, which cause complex dynamical behaviors. Moreover, including the prey refuge, we examine how our method benefits the low-density animals and affects their population dynamics.

Public Lecture

Speaker: Diego Tenoch Morales Lopez (Western)

"Adaptation reshapes the distribution of fitness effects"

Time: 09:00

Room: MC 204

Mutations drive adaptive evolution due to their heritable effects on fitness. Empirical measures of the distribution of fitness effects of new mutations (the DFE) have been increasingly successful, and have recently highlighted the fact that the DFE changes during adaptation. Here, we analyze these dynamic changes to the DFE during a simplified adaptive process: an adaptive walk across an additive fitness landscape. First, we derive analytical approximations for the fitness distributions of both available and previously fixed alleles, and use these to derive expressions for the DFE at each step of the adaptive walk. We then confirm these predictions with independent simulations that relax several simplifying assumptions made in the analysis. Along with these quantitative predictions, we find that as de novo mutations accumulate, the DFE is reshaped in two important qualitative ways: the fraction of deleterious mutations increases (a shift to the left), and the variance of the distribution decreases. Finally, our analysis makes the surprising prediction that, at least in additive fitness landscapes, adaptation may be more limited by the availability of low-fitness alleles to be replaced, rather than by the availability of beneficial mutations.