Table 1.

Summary of notations used in the paper

Global notations

N(s): number of caught schools of size s

D(s): probability that a group will be of size s

b: exponent of either N(s) or D(s)

sc: cutoff size of either N(s) or D(s)

d: spatial dimension

a, c: fitting parameters for N(s) = a s−be−(s/sc)c

General model

N: number of sites

n: number of simulated individuals

Sit: group size at site i at time t

Iit: size of group injected into site i at time t

Wkit: random hopping variable from site k to site i

D(s, t): probability that a group at time t will be of size s

Z1(ρ, t): characteristic function of D(s, t)

Z1(ρ): characteristic function of D(s)

Φ(ρ): characteristic function of Iit

〈Im〉: mth moment of Iit

f(x): crossover function of D(s)

Splitting models

p: fraction of each group splitting at each time step

N+: number of sites occupied by a group

〈s′〉: mean group size over all sites

〈s〉: mean group size over occupied sites

Psplit(s): probability of splitting for a group of size s

σ: maximum allowed group size

Model with attracting site

η: strength of attraction of the attracting site