Applications of Stochastic PDEs

To what extent should we view stochasticity as just a perturbation of a `true' deterministic PDE? In physics, such a viewpoint might have some merit, but in, for example, a single cell, stochasticity is always present and therefore we should expect the inner dynamics to work not despite the noise, but because of the noise. In these projects, we study stochastic dynamics away from the well-known deterministic patterns.

C.H.S. Hamster, P. van Heijster and E. Siero, Blurring the Busse balloon: Patterns in a stochastic Klausmeier model (2024), to appear

C.H.S. Hamster and P. van Heijster, Waves in a Stochastic Cell Motility Model (2022), Bulletin of Mathematical Biology

Stability of Stochastic Travelling waves

These papers all come out of my PhD research. The main goal was to understand travelling waves in stochastic versions of classic equations such as the Nagumo equation and the FitzHugh-Nagumo equation. We showed that stochastic travelling waves remain stable on exponentially long time scales and their average properties can approximated using perturbation methods around the deterministic wave.

M. van den Bosch, C.H.S. Hamster and H.J. Hupkes, Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems (2025), preprint

C.H.S. Hamster and H.J. Hupkes, Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations (2020), SIAM Journal on Applied Dynamical Systems

C.H.S. Hamster and H.J. Hupkes, Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise (2019), Physica D

C.H.S. Hamster and H.J. Hupkes, Stability of Travelling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise (2018), SIAM Journal on Mathemathical Analysis

C.H.S. Hamster and H.J. Hupkes, Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise (2017), SIAM Journal on Applied Dynamical Systems

Evolutionary Dynamics

There is an age-old discussion in ecology on how large ecosystems are formed and remain their stability. Our approach to this debate is to start building ecosystems from the ground up. We set up a small ecosystem model such as a resource-consumer model or Lotka-Volterra, and keep adding new species that are small perturbations of an already existing species. Hence, we mimic evolution in a computer model, which can help us understand minimum requirements for ecosystem stability.

C.H.S. Hamster, J. Schaap, P. van Heijster and J.A. Dijksman Random evolutionary dynamics in predator–prey systems yields large, clustered ecosystems (2024), Mathematical Biosciences

E. Bellavere, C.H.S. Hamster and J.A. Dijksman, Speciation in a MacArthur model predicts growth, stability, and adaptation in ecosystem dynamics(2023), Bulletin of Mathematical Biology

Thesis

Noisy Patterns