Swarm simulation using anti-Newtonian forces

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

2. Outline • Nature • Overview of observed swarming • Computer simulation • 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 • Confuses predator • Improves perception • Lowers predator success rate • Foraging • Mating • Navigation

5. Predator options • Form hunting packs • 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: • 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: • http://photography.nationalgeographic.com/