A population of dispersing beetles grows locally at rate and spreads by diffusion with

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

A population of dispersing beetles grows locally at rate

a = 0.80\,\mathrm{day^{-1}}

and spreads by diffusion with

D = 15\,\mathrm{m^2\,day^{-1}}

, so that a small density perturbation obeys

\partial_t u = au + D\partial_x^2u

. ((a)) Write down the growth rate

\sigma(q)

of a Fourier mode

e^{\sigma t + iqx}

and evaluate it for a mode of wavenumber

q = 0.20\,\mathrm{m^{-1}}

. ((b)) The patch has hostile surroundings, so the density must vanish at both of its ends,

u=0

x=0

and

x=\ell

. Find the critical patch length

\ell_{\rm c}

above which a patch of beetles grows rather than dies out.

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