1. Swarm simulation using anti-Newtonian forces
Vladimir Zhdankin
Chaos and Complex Systems
October 6, 2009

2. Outline Nature Computer simulation Overview of observed swarming
Swarming model
Selected cases
No predator
Single predator
Multiple predators
Black sheep
Conclusions

3. Swarming in nature Birds flocks Fish schools Insect swarms
Mammal herds
Human crowds

4. Why swarm? Defense from predators Foraging Mating Navigation
Confuses predator
Improves perception
Lowers predator success rate
Foraging
Mating
Navigation

5. Predator options Form hunting packs Spread and surround the swarm
Divert a prey away from swarm and catch
Spread and surround the swarm
Lead swarm into a trap

6. How does swarming happen?
Emergence
Organization arises from repetition of simple actions
Each individual makes some decisions
Chooses optimal distance from neighbors
Aligns with neighbors
Reacts to obstacles

7. Simulation Model each member as a particle (“agent”)
Model landscape as Cartesian plane
Implement force laws
Long range attraction
Short range repulsion
Friction
Anti-Newtonian force between
predator and prey

8. Anti-Newtonian force Term coined by Clint Sprott
Disobeys Newton’s Third Law
Newtonian forces are equal in magnitude and opposite in direction
Anti-Newtonian forces are equal in magnitude and equal in direction

9. Circular orbit

10. Elliptical orbit

11. Precessing orbit

12. N-body anti-Newtonian problem
With more bodies, simplest choice is to have no force between similar agents
For swarming, can add Newtonian forces
Attractive force between rabbits is natural
Force between foxes is not as obvious
Attraction to form hunting packs?
Repulsion to spread and surround rabbits?
No interaction at all?

13. Equations of motion
Can adjust to give predator repulsion instead of attraction

14. Equation parameters Agent parameters: Force parameters: Mass m
Coefficient of friction b
Priority p (scales force toward agent)
Force parameters:
Long-range force power γ
Short-range repulsion power α
Usually γ = -1 and α = -2 works best

16. Trivial case (no predator)

17. Trivial case (no predator)

18. Trivial case equilibria

19. Notes on trivial case Uninteresting approximation of nature
For complexity, add other terms:
External potential
Self-propulsion
Noise
Or, introduce a predator…

20. Single predator

21. Single predator - chaotic

22. Can the predator win?

23. Can the predator win?

24. Why not γ=0?

25. Why not γ=0?

26. Older equations of motion

27. Multiple predators

28. Two predators repelling

29. Two predators attracting

30. Predator packs

31. More complicated case

32. An unrealistic solution

33. A black sheep One swarm agent may have handicaps
Injured, sick, or weak in nature
Higher mass, friction, or priority in simulation
In nature, predators target these prey
Will it happen in simulation?

34. Black sheep - greater friction

35. Black sheep - higher priority

36. Black sheep - increased mass

37. Emergence in simulation
Swarming maneuvers
Unified motion (away from predators)
Splitting to confuse predators
Predator actions
Diverting one agent away from swarm
Capturing the black sheep
All of these come about from using the anti-Newtonian force

38. Conclusions
Swarming behavior can be approximated by modeling swarm members as particles that obey simple force laws
The anti-Newtonian force plays a critical role in the swarm dynamics
Emergence is responsible for part of Nature’s complexity

39. Acknowledgements
Clint Sprott

40. References Images of swarms in nature are from National Geographic: