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Modelling collective animal behaviour
David J. T. Sumpter
Department of Mathematics
Uppsala University
Moving together
• Aggregation in space.
• Moving in the same direction.
• Complex patterns.
Self-propelled particle models
estutu
tuvtxtx
ii
iii
)1()()1(
)1()()1( 0
current position current velocity
current velocity
position and velocity of neighbours
stochastic effect
future position
future velocity
Aggregation model in one dimension
estutu
tuvtxtx
ii
iii
)1()()1(
)1()()1( 0
current position current velocity
current velocity
position and velocity of neighbours
stochastic effect
future position
future velocity
iRj
iji
txtxsignR
s )()(1
2/,2/ e is a random number selected uniformly at random from a range
Aggregation model in one dimension
Cockroach aggregation
Cockroaches Model
Radakov’s fish
Alignment model in one dimension
estutu
tuvtxtx
ii
iii
)1()()1(
)1()()1( 0
current position current velocity
current velocity
position and velocity of neighbours
stochastic effect
future position
future velocity
e is a random number selected uniformly at random from a range 2/,2/
iRjj
i
tuR
Gs )(1
0
0
2/)1(
2/)1(
u
u
foru
foruuG
Position:
Velocity:
Tim
e
Space Time
Ave
rage
dire
ctio
n
,)()1()()1(
),()()1(
||
0
irjijii
iii
tuGtutu
tuvtxtx
Alignment model in one dimension
n
ii tu
n 1
)(1
Position:
Velocity:
Tim
e
Space Time
Ave
rage
dire
ctio
n
,)()1()()1(
),()()1(
||
0
irjijii
iii
tuGtutu
tuvtxtx
Alignment model in one dimension
n
ii tu
n 1
)(1
Position:
Velocity:
Tim
e
Space Time
Ave
rage
dire
ctio
n
,)()1()()1(
),()()1(
||
0
irjijii
iii
tuGtutu
tuvtxtx
Alignment model in one dimension
n
ii tu
n 1
)(1
Transition from disorder to order
n
ii tu
n 1
)(1 measures order in the system.
η is degree of randomness, ρ is density (number of particles over size of world).
Transition from disorder to order
n
ii tu
n 1
)(1 measures order in the system.
η is degree of randomness, ρ is density (number of particles over size of world).
Transition from disorder to order
n
ii tu
n 1
)(1 measures order in the system.
η is degree of randomness, ρ is density (number of particles over size of world).
Alignment model in two dimensions
estutu
tuvtxtx
ii
iii
)1()()1(
)1()()1( 0
current position current velocity
current velocity
position and velocity of neighbours
stochastic effect
future position
future velocity
i
i
Rjj
Rjj
t
t
es
)(sin
)(coswhere the θj are the directions of i’s neighbours
and ε is chosen uniformly at random from a range 2/,2/
.
Alignment model in two dimensions
Attraction, alignment and repulsion in three dimensions
Collective Animal Behaviour
www.collective-behavior.com
Collective Animal Behaviour
www.collective-behavior.com
Collective Animal Behaviour
www.collective-behavior.com
Collective Animal Behaviour
www.collective-behavior.com
Collective Animal Behaviour
www.collective-behavior.com
