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I have research interests in two distinct areas:
applied dynamical systems, and mathematical models of animal behaviour.
Current projects are outlined below.
Dynamical systems
Heteroclinic cycles and networks
I am interested the stability and bifurcations of heteroclinic cycles and heteroclinic networks in non-linear dynamical systems.
Such cycles are generically of high codimension but in the presence of symmetry they can be found to be robust, that is, they exist in
an open region of parameter space. Recently I have been working with Vivien Kirk (Auckland)
and Alastair Rucklidge (Leeds) investigating the notion of resonance in heteroclinic networks.
Delay equations and feedback control
The stabilisation of unstable periodic orbits using feedback control has attracted the
attention of many authors over a number of years. The time-delayed feedback method of
Pyragas has been of particular interest. Motivated by this, Mary Silber (Northwestern) and I have been
working on understanding feedback control of periodic orbits near a subcritcal Hopf
bifurcation --- a generic mechanism of creating unstable periodic orbits in nonlinear
differential equations.
Mathematical models of animal behaviour
Models of pigeon navigation
How migrating animals find their way over long distances remains one of the great, unanswered
questions facing biologists today. Despite intensive research for over 60 years, there has been no
convincing explanation of the mechanisms animals use for determining their position relative to a
target location. My research in this area combines ideas from both mathematics and behavioural
ecology, and with colleagues from the School of Biology Sciences, I hope to develop a new mathematical model of animal navigation. I have recently been awarded a Marsden Fast Start grant for this research.
Due to their ease of handling and willingness to home, homing pigeons have long been the
experimental model for the study of animal navigation. Our research will involve the development
of geometric techniques used to explain an observed ‘orientation error’ in the initial homing
directions of pigeons. We will then develop a predictive mathematical model for how animals
navigate over long distances. These results will be applicable to a wide variety of migratory species.
We expect that our results will explain how birds such as godwits can fly non-stop from Alaska to
New Zealand, which requires locating a target only 2-3 degrees wide when migration begins.
Weakly electric fish
In collaboration with Malcolm MacIver and others at Northwestern University, I have been investigating
the interaction of body shape and movement capability in weakly electric fish.
Animal behaviour arises through a complex mixture of biomechanical,
neuronal, sensory, and control constraints. By focusing on a simple,
stereotyped movement, the prey capture strike of a weakly
electric fish, we show that the trajectory of a strike is one which minimises
effort.
Specifically, we model the fish as a rigid ellipsoid moving through a fluid with
no viscosity, governed by Kirchhoff's equations. This formulation
allows us to exploit methods of discrete mechanics and optimal control
to compute idealized fish trajectories that minimize
a cost function. We compare these to measured prey capture strikes
of weakly electric fish from a previous study.
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