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CATEGORIES:Applied Mathematics Seminar Series
SUMMARY:Developing PDE-compartment hybrid frameworks for m
odeling cell migration - Dr Christian Yates\, Univ
ersity of Bath
DTSTART:20151109T140000Z
DTEND:20151109T150000Z
UID:TALK1865AT
URL:/talk/index/1865
DESCRIPTION:Spatial reaction-diffusion models have been employ
ed to describe many emergent phenomena in biologic
al systems. The modelling technique most commonly
adopted in the literature implements systems of pa
rtial differential equations (PDEs)\, which assume
s there are sufficient densities of particles that
a continuum approximation is valid. However\, due
to recent advances in computational power\, the s
imulation\, and therefore postulation\, of computa
tionally intensive individual-based models has bec
ome a popular way to investigate the effects of no
ise in reaction-diffusion systems in which regions
of low copy numbers exist.\n\nThe specific stocha
stic models with which we shall be concerned in th
is talk are referred to as `compartment-based' or
`on-lattice'. These models are characterised by a
discretisation of the computational domain into a
grid/lattice of `compartments'. Within each compar
tment particles are assumed to be well-mixed and a
re permitted to react with other particles within
their compartment or to transfer between neighbour
ing compartments. \n\nIndividual-based stochastic
models provide microscopic/mesoscopic accuracy but
at the cost of significant computational resource
s. Models which have regions of both low and high
concentrations often necessitate coupled macroscal
e and microscale modelling paradigms. This is beca
use microscale models are not feasible to simulate
at large concentrations and macroscale models are
often inappropriate at small concentrations.\n\nI
n this work we develop two hybrid algorithms in wh
ich a PDE in one region of the domain is coupled t
o a compartment-based model in the other. Rather t
han attempting to balance average fluxes\, our alg
orithms answer a more fundamental question: `how a
re individual particles transported between the va
stly different model descriptions?' First\, we pre
sent an algorithm derived by carefully re-defining
the continuous PDE concentration as a probability
distribution. Whilst this first algorithm shows v
ery strong convergence to analytic solutions of te
st problems\, it can be cumbersome to simulate. Ou
r second algorithm is a simplified and more effici
ent implementation of the first\, it is derived in
the continuum limit over the PDE region alone. We
test our hybrid methods for functionality and acc
uracy in a variety of different scenarios by compa
ring the averaged simulations to analytic solution
s of PDEs for mean concentrations.\n
LOCATION:Aston Webb WG12
CONTACT:David Smith
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