Modelling collective animal behaviour David J. T. Sumpter Department of Mathematics Uppsala University

Modelling collective animal behaviour 1.Coming together: models of animal aggregation. 2.Moving together: models of group movement. 3.Collective motion of animal groups: experiment and theory (Saturday)

Coming together Why join a group? How groups form? What group size distributions should we expect?

Why join a group? Natural selection acts on selfish individuals Can conserve energy by being near others. Someone else will get eaten if a predator attacks. Can eat the food that others find. Can build structures together which can’t be built alone.

Why join a group?

Empirical examples Survival of spidersStress in lemurs

Optimal vs Evolutionarily Stable?

The ’tradegy’ is that groups will become a size where there is no longer any benefit to being in a group! Such tradegies are common outcomes in game theoretic models of evolution. Prisoners dilemna, public goods games etc.

Simulating stable group size Consider f(n) = n exp (n/10) and an environment with s=2000 available sites. Assume initially that at half the sites, i=1 to 1000, the number of individuals at the site, n i (0), is drawn from a uniform distribution with minimum 10-r and maximum 10+r. The other half of the sites, i=1001 to 2000, are unoccupied, i.e. n i (0)=0. On each time step t a random individual is picked. It then calculates the fitness function for all of the sites were it to move to that site, i.e. f(n j (t)+1) for all sites apart from the site i that is already at. If f(n j (t)+1)> f(n i (t)) for some j then the individual moves to the site which has the maximum value of f(n j (t)+1). If more than one site has the same value of f(nj(t)+1) then one of these sites is picked at random. The above process is continued until no further moves are possible.

Simulating stable group size

Stable groups are larger than optimal size, but smaller than Sibly size. Very narrow group size distribution.

What do real group size distributions look like? Spider colony sizes

What do real group size distributions look like? Buffalo herd sizes

What do real group size distributions look like? Tuna fish catch distributions

Power laws Group size n occurs with frequency. Equivalently Thus the slope in the log-log plot gives the exponent of the power law.

Where do power law distributions come from? Combinations of exponential distributions. Inverses of quantities. Random walks. Phase transitions and critical phenomena. Preferential attachment See Newman (2005) or Sornette (2003)

Preferential Attachment