### Dynamical systems + PDEs overview:

I have been investigating noise-induced tipping in a collaborative project using numerical continuation techniques. In a different direction, I am building PDE models of zebrafish patterning alongside my agent-based work; by conducting a bifurcation analysis in this setting, I hope to shed light on zebrafish pattern mutants and more closely relate analytically-tractable PDE approaches with underlyingly discrete biological dynamics.

### Noise-induced tipping in periodically forced systems

Together with Yuxin Chen, John Gemmer, and Mary Silber, I am interested in how additive noise impacts periodically-forced systems with a double-well potential structure. While our original motivation was based on seasonal fluctuations in the depth of Arctic sea ice (in this setting, tipping could represent jumping from a stable ice-covered ocean to a stable ice-free Arctic), we have focused on a more general system that allows us to explore different time scales in the problem:

dXt = (1/ε) ( Xt - Xt3 + α + A cos(2πt) )dt + σdWt = (1/ε) f(x,t) + σdWt,
where σ is the noise strength and ε measures how quickly the flow relaxes to its steady states relative to the external forcing period. The case when ε << 1 has been studied extensively, so we instead focus on ε = O(1). We use a path integral formulation to identify most likely tipping paths between stable periodic orbits, and my role has been to explore bifurcations in the problem. Interestingly, the associated Euler-Lagrange equations have a Hamiltonian system, namely:

x' = ψ + (1/ε) f(x,t)
ψ' = (1/ε) fx(x,t) ψ + ( σ2/(2ε) ) fxx(x,t),
where ψ = x' - (1/ε) f(x,t) is the "momentum" and measures the deviation from the deterministic dynamics introduced by noise. The Hamiltonian system admits several periodic orbits, some of which correspond to the solutions of the underlying deterministic problem when $\sigma = 0$. I have been using AUTO-07p (numerical continuation) to explore bifurcations in the Hamiltonian equations, which may be related to regions with qualitatively different noise-induced tipping behavior.

Collaborators:
Yuxin Chen (Northwestern University), John Gemmer (Wake Forest University), Mary Silber (University of Chicago)

Publications:

### Non-local PDE models of zebrafish patterning

While agent-based models are often closely related to the underlying application, these stochastic, rule-based systems are challenging to analyze. In contrast, continuum limits lend themselves to analysis. To combine the benefits of both approaches, we are developing PDE models of zebrafish patterning based on my previous agent-based models. Because pigment cell interactions on the zebrafish skin occur at both short and long range, our models take the form of non-local conservations laws (or aggregation equations) and feature non-local reaction terms (modeled using convolution terms).

Collaborators:
José A. Carrillo (Imperial College London), Shigeru Kondo (Graduate School of Frontier Biosciences, Osaka University), Markus Schmidtchen (Imperial College London), Chandrasekhar Venkataraman (University of Sussex)

Alexandria Volkening
Last updated July 7, 2019