Title:  One Step Too Far: How Discrete Dynamics Predict Thresholds of Sudden Collapse, from Cells to Cosmos.

Time:  Saturday,  2:00 PM - 3:00 PM

Abstract: Discrete dynamical systems offer a simple yet powerful way to understand how complex systems evolve through step-by-step change. In this talk, we explore how such models reveal critical thresholds that separate survival from sudden collapse.

Focusing on the Allee effect, we show how populations can fail when their size reaches a critical level, even in environments that would otherwise support growth. Building on a discrete-time epidemic framework, we demonstrate how disease dynamics can drive a population past this threshold, leading to the extinction of both the host and the disease itself. These results highlight how small changes in parameters or initial conditions can produce dramatically different long-term outcomes.

Through visual simulations and minimal technical language, we extend these ideas to a broader context, including models of cancer cell dynamics, conflict scenarios, and large-scale systems. Across these examples, a common theme emerges: discrete dynamics naturally encode thresholds that govern persistence or collapse.

This talk emphasizes how simple iterative rules can capture deep, sometimes counterintuitive behavior, offering insight into systems ranging from biological populations to processes at much larger scales.

Bio: Sam Habach is an Assistant Professor of Mathematics at the Massachusetts Maritime Academy. He earned his PhD in Applied Mathematics from the University of Rhode Island, where his research focused on the stability and global dynamics of discrete population models, including effects such as cooperation, harvesting, and threshold-driven behavior. His work lies in the area of discrete dynamical systems, with applications ranging from epidemiology to complex biological and physical systems. He has presented his research at several American Mathematical Society

meetings and has a background in both mathematics and physics education. He is particularly interested in developing intuitive and visual approaches to mathematical modeling that make complex ideas accessible to broader audiences.