Research

My interests lie at the intersection of nonlinear dynamics, stochastic analysis, and mathematical modeling. In my graduate studies, I worked on functional differential equations, studying growth rates and blow-up of solutions in deterministic and stochastic systems with memory. More recently, I have focused on applying mathematical methods to build and analyze models in applications and I have ongoing interdisciplinary collaborations with colleagues in ecology, developmental biology, and epidemiology. Mathematically, my applied projects involve studying the dynamics of stochastic particle systems, and partial and integro-differential equations.

Collaborators:

John Appleby (Dublin City University, PhD Advisor)
Lauren Childs (Virginia Tech)
Shen-Ju Chou (Academia Sinica)
Christina Edholm (Scripps College)
Simon Levin (Princeton University, Postdoctoral Mentor)
Joan Ponce (Arizona State University)
Olivia Prosper (University of Tennessee Knoxville)
Zhuolin Qu (University of Texas San Antonio)
Carla Staver (Yale University)
Jonathan Touboul (Brandeis University, Postdoctoral Mentor)
Jin Wang (Stony Brook)
Li Xu (Chinese Academy of Sciences)
Lihong Zhao (University of California Merced)

Mathematical Ecology: Vegetation Models

A thorough understanding of the dynamics and stability of tropical biomes such as rainforest and savanna is crucial to conservation efforts. I am presently working on rigorously deriving and analyzing spatially extended versions of classical models in this domain – our models aim to more realistically reflect vegetation dynamics by accounting for the spatial structure and environmental heterogeneity observed empirically. Another key aim of this work is to quantify the stability of these ecosystems to noise and parameter shifts.

A1-A4: Comparison of waves of invasion in stochastic model versus solution of the integro-differential equations (IDEs) governing the law of the stochastic model. B1-B2: Front pinning in an interacting particle system versus solution of governing IDEs (heterogeneous medium). B3: Profile of the solutions overlaid with bifurcation diagram of the non-spatial system.

L. Xu, D. D. Patterson, S. A. Levin, J. Wang, New early-warning signals for critical transitions in ecological systems, Proceedings of the National Academy of Sciences, Vol. 120, No. 5 (2023), e2218663120.

L. Xu, D. D. Patterson, A. C. Staver, S. A. Levin, J. Wang, Unifying deterministic and stochastic ecological dynamics via a landscape-flux approach, Proceedings of the National Academy of Sciences, Vol. 118, No. 24 (2021), e2103779118.

D. D. Patterson, A. C. Staver, S. A. Levin, J. D. Touboul, Probabilistic foundations of spatial mean-field models in ecology and applications, SIAM Journal on Applied Dynamical Systems, Vol. 19, No. 4 (2020), 2682-2719.

Mathematical Modeling: Developmental Biology

Working closely with experimental collaborators, we developed a novel mathematical model to explain the unexpected formation of ectopic cortical regions in the brains of mutant mice during early development.

2D_phenotypes_image — Comparison of the predicted brain phenotypes from our mathematical model (row 1) with experimentally observed phenotypes from 5 different species of mutant mice (row 2).

J. Feng*, W. H. Hsu*, D. D. Patterson, C. S. Tseng, Z. H. Zhuang, H. W. Hsin, Y.T. Huang, A. Faedo, J. L. Rubenstein, J. D. Touboul and S.J. Chou, COUP-TFI specifies the medial entorhinal cortex identity and induces differential cell adhesion to determine the integrity of its boundary with neocortex, Science Advances, Vol. 7, No. 27 (2021), eabf6808. [*equal contribution]

Mathematical Epidemiology

Malaria is a mosquito-borne disease that causes hundreds of thousands of deaths each year. Individuals acquire immunity, of various types, from both exposures over time and by vaccination. Since malarial impacts vary significantly by age, we constructed and analyzed a mathematical model with age structure to assess the impact of immune feedback on the disease dynamics.

solutions_summary_plot — Left: Stable age distribution based on fit to Kenyan demographic data. Center/right: Heatmap of the proportion of infected people at a given age versus mosquito infectivity.

Z. Qu*, D. D. Patterson*, L. Childs, C. Edholm, J. Ponce, O. Prosper and L. Zhao, Modeling immunity to malaria with an age-structured PDE framework, SIAM Journal on Applied Mathematics (2023), accepted. [*equal contribution]

Functional Differential Equations

Finite-time blow-up of systems with memory

There is a vast literature on blow-up criteria for nonlinear Volterra equations and also a considerable array of results regarding the estimation of the blow-up time. However, the problem of determining the behavior of solutions near blow-up is relatively open and we address this issue for some classes of nonlinear functional differential equations.

J. A. D. Appleby and D. D. Patterson, Blow-up and superexponential growth in superlinear Volterra equations, Discrete and Continuous Dynamical Systems (Series A), Vol. 38, No. 8 (2018), 3993-4017.

J. A. D. Appleby and D. D. Patterson, Growth rates of solutions of superlinear ordinary differential equations, Applied Mathematics Letters, Vol. 71 (2017), 30-37.

Discrete systems with memory

Volterra summation equations are general discrete-time models for processes whose evolution depends on their entire history (e.g. time series models in economics and finance). If solutions become unbounded it is natural to rescale them, but does this process preserve economically relevant properties? We answer this question for linear equations with both random and deterministic forcing sequences.

J. A. D. Appleby and D. D. Patterson, Large Fluctuations and growth rates of linear Volterra summation equations, Journal of Difference Equations and Applications, Vol. 23, No. 6 (2017), 1047-1080.

Growth rates of systems with memory

The evolution of many phenomena depends not only on their present state but also on their past states. In continuous-time, incorporating this dependence leads to the study of functional differential equations (FDEs). These papers investigate how the memory of past states affects the growth rate of such systems, and how dynamics are impacted by random and deterministic forcing.

stochastic_bounds — Simulations illustrating the sharpness of almost sure asymptotic bounds on solutions for nonlinear stochastic FDEs developed in the works below (see Appleby and Patterson, AMC (2021) for details).

J. A. D. Appleby and D. D. Patterson, Growth and fluctuation in perturbed nonlinear Volterra equations, Applied Mathematics and Computation, Vol. 396, (2021) 125938.

J. A. D. Appleby and D. D. Patterson, Growth rates of sublinear functional and Volterra differential equations, SIAM Journal on Mathematical Analysis, Vol. 50, No. 2 (2018), 2086-2110.

J. A. D. Appleby and D. D. Patterson, Memory dependent growth in sublinear Volterra differential equations, Journal of Integral Equations and Applications, Vol. 29, No. 4 (2017), 531-584.

J. A. D. Appleby and D. D. Patterson, Hartman-Wintner growth results for sublinear functional differential equations, Electronic Journal of Differential Equations, Vol. 2017, No. 21 (2017), 1-45.

Convergence rates of stable solutions

How quickly do solutions to a nonlinear ordinary differential equation with a globally stable equilibrium decay? These papers consider this question for scalar ordinary and stochastic differential equations. The use of the theory of regularly varying functions is a recent theme in QTDE and we employ regular variation here to prove very precise asymptotic results.

J. A. D. Appleby and D. D. Patterson, Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity, Electron. J. Qual. Theory Differ. Equ., Proc. 10th Coll. Qualitative Theory of Diff. Equ., No. 3 (2016), 1-38.

J. A. D. Appleby and D. D. Patterson, On necessary and sufficient conditions for preserving convergence rates to equilibrium in deterministically and stochastically perturbed differential equations with regularly varying nonlinearity, Recent Advances in Delay Differential and Difference Equations, Springer Proceedings in Mathematics & Statistics 94 (2014), 1-85.