As a mathematical biologist who specializes in modeling phenomena with differential equations, Dr. Ryan’s studies how complex biological systems organize themselves. This general topic covers things ranging from how colonies of bacteria interact in suspension to how groups of insects move in swarms to avoid predators.
Dr. Ryan highlights numerous “big picture” ideas in mathematical modeling, which he broadly splits into 3 parts: modeling, simulation, & analysis. Modeling to the act of writing down (differential) equations to capture the essential features of the physical system in question. The key here is “making the model as simple as possible, but no simpler”. Dr. Ryan considers this his favorite part, and his particular strong suite. A constant question is that of parameter estimate to ensure the terms in the model are realistic.
Once a model exists then simulation and analysis come into play. Using the tools of analysis one can work directly with the mathematical equations hoping to prove things like existence of a solution and solution uniqueness. Here model complexity matters, as a complex model may be analytically intractable, meaning it’s impossible to say much about the model using pure math.
Simulation goes the other direction from analysis. Rather than work with the differential equations, the equations are somehow discretized into a form digestible to computers, and the research can then simulate the system directly. Here there are challenges such as stability and computational efficiency. When a given model is discetized, it may be that a small change in the parameters results in a major changes in the output. The simulation is the unstable and may not be trustworthy. For computational efficiency, the actual details of how the model is programmed matter. Here Dr. Ryan highlights tricks he uses such as GPU programming that also reduces the communication cost between GPUs during a simulation.
Overall Dr. Ryan delivers a masterful overview of major aspects of mathematical modeling covering broad principles as well as specific examples from his own work.