Differential and difference equations play a crucial role in the modelling of problems in the natural and social sciences, as well as being of intrinsic mathematical interest. However, in almost all cases, we cannot solve such equations explicitly and must answer questions about the general properties of solutions without formulae for them. For example, do solutions always stay positive, do they decay or grow over time, do they remain bounded? This area of research is typically referred to as the qualitative theory of differential equations (QTDE). I am particularly interested in nonlinear equations, those which incorporate nonlocal effects (in time or space), and stochastic equations – all of these effects play a role in the modeling of real-world systems, and generate interesting and difficult mathematical problems.

Mathematical models of embryonic development

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.

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, J. D. Touboul and S.J. Chou, COUP-TFI specifies entorhinal cortex and determines the location and integrity of its border through cell affinity mechanisms, submitted.

Spatial vegetation models in mathematical ecology

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.

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.

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, accepted.

Finite-time blow-up of systems with memory

There is a rich 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 this is the most novel aspect of my work in this area.

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. These papers concern how the memory of past states affects the growth rate of such systems.

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.