1. Introduction to game dynamics Pierre Auger IRD UR Geodes, Centre d’île de France et Institut Systèmes Complexes, ENS Lyon

2. Summary z Hawk-dove game z Generalized replicator equations z Rock-cissor-paper game z Hawk-dove-retaliator and hawk-dove-bully z Bi-matrix games

3. Modelling aggressiveness

4. Fighting for resources Dominique Allainé, Lyon 1

5. Hawk-Dove game z Payoff matrix z Gain z Cost

6. Playing against a population z Hawk reward z Dove reward z Average reward

7. Replicator equations With

8. Replicator equations Because Leading to then

9. Hawk-dove phase portraits

10. Replicator equations z G<C, dimorphic equilibrium J. Hofbauer & K. Sigmund, 1988 z G>C, pure hawk equilibrium Butterflies

11. Replicator equations : n tactics (n>2) z Payoff matrix z a ij reward when playing i against j

12. Replicator equations With z Average reward z Reward player i

13. Equilibrium With z Unique interior equilibrium (linear) z Corner

14. Rock-Scissor-Paper game z Payoff matrix

15. Replicator equations

16. Four equilibrium points z Unique interior equilibrium

17. Replicator equations

18. Local stability analysis z Unique interior equilibrium saddle center

19. zLinear 2D systems (hyperbolic)

20. R-C-P phase portrait z First integral

21. Hawk-Dove-Retaliator game z Payoff matrix

22. H-D-R phase portrait

23. Hawk-Dove-Bully game z Payoff matrix

24. H-D-B phase portrait

25. Bimatrix games (two populations) z Pop 1 against pop 2 z Pop 2 against pop 1

26. Bimatrix games (2 tactics) z Average reward z Reward player i

27. Adding any column of constant terms z Pop 1 against pop 2 z Pop 2 against pop 1

28. Replicator equations

29. Five equilibrium points z Unique interior equilibrium (possibility)

30. Jacobian matrix at (x*,y*)

31. Local stability analysis z Unique interior equilibrium (trJ=0 ; center, saddle) z Corners (Stable or unstable nodes, saddle)

32. zLinear 2D systems (hyperbolic)

33. Battle of the sexes z Females : Fast (Fa) or coy (Co) z Males : Faithful (F) or Unfaithful (UF)

34. Battle of the sexes z Males against females

35. Battle of the sexes z Females against males

36. Adding C/2-G in second column

37. Replicator equations

38. Five equilibrium points z Unique interior equilibrium : C<G<T+C/2

39. Local stability analysis (center) z Existence of a first integral H(x,y) :

40. Phase portrait (existence of periodic solutions)