On a reef, cleaner fish visit client fish along an interaction network that behaves…

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

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

On a reef, cleaner fish visit client fish along an interaction network that behaves as an Erdos-Renyi random graph with mean degree

\langle k\rangle = 8

. Modelling each fish's daily activity rhythm as a phase oscillator, the coupling needed for a fully connected shoal (all-to-all) to entrain is

K_0 = 6.4\,\mathrm{rad\,day^{-1}}

. ((a)) Estimate the coupling

K_{\rm c}

at which the reef network itself begins to entrain, using the leading eigenvalue of its adjacency matrix. Use

\kappa_1\approx\langle k\rangle+1

for an Erdos-Renyi graph and

K_{\rm c}\approx K_0/\kappa_1

. ((b)) A red-tide event kills fish at random across the reef. Using the site-percolation threshold for random failure, find the fraction of fish that must be lost before the interaction network's giant component disappears.

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