In Preparation

1. Timothy V Roberts, Paul Carter, Karl Glasner. Existence of amphiphilic bilayer and micelle equilibria in a density functional model. Submission expected in June.

2. Timothy V Roberts, Punit Gandhi, Mary Silber. Curvature and confinement: dryland vegetation patterns on undulating, sloped terrain.

3. Tina Wang, Punit Gandhi, Timothy V Roberts, Mary Silber. Wavenumber selection for a random kick-flow model of dryland vegetation pattern formation.

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Publications

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1. Timothy V Roberts and Björn Sandstede. Homoclinic snaking of contact defects in reaction- diffusion equations. Nonlinearity, 38(11):115001, November 2025.

2. Kristen E. Harley, Peter van Heijster, Robert Marangell, Graeme J. Pettet, Timothy V. Roberts,and Martin Wechselberger. (In)stability of Travelling Waves in a Model of Haptotaxis. SIAM Journal on Applied Mathematics, 80(4):1629–1653, January 2020.

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Current Projects

• Curvature and confinement: dryland vegetation patterns on undulating, sloped terrain.
  
  Vegetation in arid environments must overcome scarcity of resources, primarily of water, to survive. Yet we find vegetation in very arid environments all over the world. Early phenomenological models by Klausmeier demonstrated that vegetation on sloped terrain is able to survive in arid conditions by breaking up into densely vegetated stripes along the elevation contours of the slope. Bare soil regions between the stripes collect water which flows downhill before being caught and captured by the vegetation stripe immediately following. The formation of vegetation patterns can then be seen as a mechanism for developing resilience against aridity.
  
  While there are multiple models of this type in use today, they are typically studied on idealized 1-dimensional slopes. In this work, I study the formation of patterns on 2-dimensional undulating slopes using a combination of numerical methods, numerical continuation and spectral theory. My results investigate links between the topography and the selected patterns, and how the formation of patterns affect the resilience of vegetation as water becomes more scarce. This is a joint work with Punit Gandhi and Mary Silber. Work is in preparation.
  
  Keywords: 2-d patterns, pattern formation, spectral problems, pattern resilience
   

• Existence of amphiphilic bilayer and micelle equilibria in a density functional model
  
  I study the existence of radially symmetric patterns in the formation of amphiphilic copolymers. Amphiphilic copolymers have properties of both hydrophobic and hydrophilic chemicals, and are self-assembling (ie. relatively easy to produce). As such, they have many industrial and medical uses. Importantly, their properties depend on the arrangement of their constituent parts and so there is interest in predicting their geometry from system parameters at onset of assembly, such as their width or radius. 
  
  Solutions to these models with radial symmetry in 1, 2 and 3 dimensions, correspond to stationary arrangements of different types: planar sheets,  cylindrical rods and spherical balls or shells, respectively. From a mathematical point of view the 1-dimensional, and the 2 or more dimensional cases present different challenges. In 1-dimension, the equations are simple, but more effort is required to find width of the planar structures produced. In 2 and 3 dimensions, you have to contend with a radial laplacian which is non-autonomous and singular at the origin.
  
  I have used careful geometric singular perturbation theory arguments (GSPT), including the exchange lemma and a three lengthscale (timescale) approach, to prove the existence of radially symmetric structures and to obtain leading order expressions for the pattern radius in terms of system parameters. Joint work with Paul Carter and Karl Glasner, results in preparation.
  
  The framework used to prove these existence results provides a promising way to approach rigorous existence and stability of vegetation spots in 2 spatial dimensions. Past numerical investigations in the literature shows that such patterns should only be stable when they have a small enough radius, but no rigorous results are currently known. This is ongoing work with Paul Carter.
  
  Keywords: GSPT, pattern formation, 2-d patterns, radial solutions
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Past Projects

• Snaking bifurcations in the Brusselator
  
  I study the existence of defect solutions (1-dimensional spiral and target waves) in reaction diffusion equations. Using geometric methods I have shown that defect solutions exist under broad assumptions and that these solutions undergo a version of homoclinic snaking bifurcation. By obtaining, qualitative and quantitative descriptions of the resulting bifurcation diagrams, I have proved the existence of previously unknown patterns to reaction-diffusion equations: asymmetric contact defects with arbitrary phase offsets. I have also shown that these new patterns are an essential feature of snaking that come from reconciling the dynamical systems properties of contact defects, and the known properties of defects from pattern formation. Joint work with Bjorn Sandstede. Results published in [1].
  
  Keywords: spatial dynamical systems, pattern formation, reaction diffusion equations
   

• Spectral stability of fronts via the Riccati-Evans function 
  
  Spectral stability of traveling fronts in reaction-advection-diffusion equations using new geometrically inspired tools. We cast the eigenvalue problem of traveling fronts as a related nonlinear problem on the Grassmannian manifold. Through this, I have implemented a simple and efficient method for numerically computing the location of point spectra for traveling waves. Joint work with Kristen E. Harley, Peter van Heijster, Robert Marangell, Graeme J. Pettet and Martin Wechselberger. Results published in [2].
  
  Keywords: Spectral problem, Riccati methods
   

• Thermoregulation in a model for PO/AH neurons
  
  The pre optic area and anterior hypothalamus (PO/AH) is thought to be an integrative center of thermoregulation in the body, ie. it is responsible for responding to changes in body temperature and environment to stop the body overheating or overcooling. I studied a Hodgkin-Huxley style model for PO/AH neurons to determine how their behavior changes with temperature. I used numerical bifurcation techniques and explained the results GSPT.
  
  Keywords: GSPT, mathematical physiology​​​​​​