An ecological survey models a network of forest patches linked by wildlife corridors…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Random graphs, the giant component and percolation on networks

Problem

An ecological survey models a network of forest patches linked by wildlife corridors as a configuration model: half of the patches have exactly

k=3

corridors and half have exactly

k=5

corridors, with corridor endpoints matched uniformly at random subject to these degrees, giving a locally tree-like network in the sparse limit. Random habitat loss removes each patch independently with probability

1-\phi

, leaving a fraction

\phi

of patches (site percolation on the network). (a) Starting from the self-consistency equation

u = 1-\phi+\phi G_1(u)

for the probability

u

that a neighbour reached along a corridor fails to lead into the giant surviving component, derive the percolation threshold

\phi_{\rm c} = \frac{\langle k\rangle}{\langle k^2\rangle - \langle k\rangle}

by linearising the map near

u=1

and demanding that the giant component just fails to exist. (b) Evaluate

\langle k\rangle

\langle k^2\rangle

and

\phi_{\rm c}

for the stated corridor network. (c) Compare

\phi_{\rm c}

with the site-percolation threshold

p_{\rm c} = 0.5927

of the square lattice, held from Computational Physics, where every site has exactly four nearest neighbours, the corridor network's mean degree, and explain why the two thresholds differ so much.

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