Math Biology

My interests in mathematical biology span a number of fields. More recently I have become interested in the interaction between fluid motion and microbial or animal behaviour. This includes the behaviour of gyrotatic organisms with complex fluid motions and mathematical modelling of flocking behaviour. I have also been involved in mathematical modelling of prehistoric population dynamics, motivated by the spread of farming in the Neolithic epoch. More information about each topic can be found below.

bird.jpg

Collective motion in biological systems

Bird flocks and fish schools are spectacular natural displays, I am interested in trying to capture such behaviour through mathematical modelling.

One of the simplest models of flocks or swarms is the Vicsek model, through which flocking behaviour is an emergent phenomena. Here each organism is modelled as a point object, which moves according to

(1)
\begin{align} \frac{d \mathbf{x}_i}{d t}=\mathbf{v}_i, \end{align}
(2)
\begin{align} \mathbf{v}_i=V(\cos\theta_i\sin\phi_i,\sin\theta_i\sin\phi_i,\cos\phi_i). \end{align}

Flocking emerges if every $\tau_f$ time units a particles swimming/flying direction is updated based on neighbouring particles within a critical radius $R_c$,

(3)
\begin{align} \theta_i=\langle \theta \rangle_{R_c}+\epsilon, \:\: \phi_i=\langle\phi\rangle_{R_c}+\epsilon, \end{align}

where $\epsilon$ is normally distributed noise. Below shows a numerical simualtion of this system with 20000 particles ($R_c=0.03, \epsilon \sim {\cal N}(0,0.1^2)$). Note the emergence of coherent 'flocks'.

flock.gif

I am currently investigating the role of external forcing on the morphology of flocks using a modified form of the Vicsek model. A first publication in this direction has been published in Phys. Rev. E, a preprint is available on the arXiv.

Swimming micro-organisms

In collaboration with Nick Hill (Glasgow) and a PhD student Scott Richardson, I have started to work on modelling gyrotatic cells in vortical flows. Gyrotactic cells are bottom heavy and experience a gravitational torque which acts to re-orientate them if they move away from their preferred vertical orientation. Consequently cells focus into downwelling regions resulting in the formation of plumes. Irrespective of shape, orientation can also be affected by local vorticity in the fluid, non-spherical cells will also experience an additional torque due to strain in the flow. Pedley & Kessler (Annu. Rev. Fluid Mech. 24, 313, 1992) derived the following system of ODEs by considering a balance of these torques,

(4)
\begin{align} \frac{d\mathbf{p}}{dt}=\dfrac{1}{2\Psi} [ \mathbf{k} -(\mathbf{k} \cdot \mathbf{p}) \mathbf{p}]+\frac{1}{2}\boldsymbol{\omega} \times \mathbf{p} + \alpha (\mathbf{I}-\mathbf{p}\mathbf{p})\mathbf{E} \mathbf{p} \end{align}
(5)
\begin{align} \frac{d\mathbf{x}}{dt}=\mathbf{u}+\Phi \mathbf{p} \end{align}

Here $\mathbf{p}$ is the organisms orientation (i.e. direction of self-propelled motion), $\mathbf{x}$ is the position vector, $\mathbf{u}=\mathbf{u}(x,t)$ is the background flow-field with $\boldsymbol{\omega}=\nabla \times \mathbf{u}$ the vorticity. $\mathbf{k}$ is the preferred swimming direction (we always take this to be upwards i.e. $\mathbf{k} =(0,0,1)^{\rm T}$ and $\Psi$ is a dimensionless parameter which measures the stability of the cells orientation, such that if $\Psi \omega>1$ the cell can be overturned by vorticity. $\Phi$ measures the cells swimming speed relative to the background flow speed. The final term in the equation for $\dot{\mathbf{p}}$ arises from assuming the cells are prolate ellipsoids with $\alpha=(\gamma^2-1)/(\gamma^2+1)$ where $\gamma$ is the ratio of the cell's major to minor axes ($\mathbf{E}$ is the rate of strain tensor and $\mathbf{I}$ is the identity matrix). We have started to investigate particles subject to these equations of motion in 3D chaotic flows. The figure below shows the position of gyrotactic 'particles' in the ABC flow with (from top to bottom) $\alpha=0,\,2/3,\, 1$. Note the appearance of filamentary structures when $\alpha=2/3$ showing the affect of strain in the flow, which appear to cancel with vorticity when $\alpha=1$.

ABC_gyrotaxis.png

Prehistoric population dynamics

I have also been involved in using reaction-diffusion models to simulate the spread of Neolithic farmers, from the near-East into Europe. This work was in collaboration with Graeme Sarson, Anvar Shukurov, Richard Boys, Andrew Golightly and Daniel Henderson.

Following previous work we numerically solved the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation with an advective velocity employed on coastlines and rivers,

(6)
\begin{align} \frac{\partial N(\mathbf{x},t)}{\partial t}+\nabla \cdot (\mathbf{V}N)=\gamma N\left(1-\frac{N}{K}\right)+ \nabla \cdot (\nu \nabla N), \end{align}

on a 2D mesh, in spherical coordinates.
Here $\gamma$ is the growth-rate of the population, which is estimated from the generation time of the population, and is fixed in our simulations.
Both $K$ (the carrying capacity) and $\nu$ vary spatially (according to altitude) and are calculated from topographical data available from the National Geophysical Data Center. This work has so far resulted in three publications:

  • D.A. Henderson, A.W. Baggaley, A. Shukurov, R.J. Boys, G.R. Sarson & A. Golightly, Regional variations in the European Neolithic dispersal: role of the coastlines, to appear in Antiquity.
  • A.W. Baggaley, R.J. Boys, G.R. Sarson, A. Golightly & A. Shukurov, Inference for a reaction-diffusion model of population dynamics in the Neolithic period, Ann. Appl. Stat., 6, 1352-1376, (2012)
  • A.W. Baggaley, G.R. Sarson, A. Shukurov, R.J. Boys & A. Golightly, Bayesian inference for a wavefront model of the Neolithisation of Europe, Phys. Rev. E, 86, 016105, (2012)

PDF copies of the articles can be found on my publications page. In the future we plan to investigate agent based models of the spread.

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