Research

My research lies at the interface of nonlinear dynamics, stochastic analysis, and mathematical modelling. My graduate work examined qualitative properties of nonlinear differential systems with memory and subject to random noise, with a focus on growth rates and blow-up of solutions in both deterministic and stochastic systems. My postdoctoral and subsequent research has applied mathematical methods — including dynamical systems, stochastic particle systems, and partial and integro-differential equations — to interdisciplinary collaborations in ecology, developmental biology, and epidemiology. See the Publications page for a full list of papers.

Mathematical Ecology

My ecology research focuses on spatial models of vegetation dynamics (especially forest-savanna ecosystems), alternative stable states, and critical transitions (abrupt regime shifts). Central themes are understanding how spatial heterogeneity and noise interact to shape the stability landscapes of ecosystems, and developing early-warning indicators for impending transitions.

A1–A4: Comparison of waves of invasion in the stochastic model versus the 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: Solutions overlaid with bifurcation diagram of the non-spatial system.

Mathematical Epidemiology: Malaria

I work on mathematical modelling of malaria at both the population and within-host scales. Using age-structured PDE frameworks, my collaborators and I modelled the accumulation of anti-malaria immunity in humans and its consequences for parasite evolution and vaccine design. In terms of within-host dynamics of malaria, I have worked on the question of how immune pressures shape the evolution of parasite transmission investment.

Left: Optimal nonconstant and constant parasite investment strategies from our within-host mathematical model with immune response. Right: Comparison of temporal host infectiousness profiles for the optimal strategies.

Functional Differential Equations

My doctoral research studied the qualitative behaviour of Volterra integral and functional differential equations, including the growth and blow-up of solutions in superlinear systems, large fluctuations in stochastic systems, and convergence rates to equilibrium.

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).