Local Replicator Dynamics

1. Local Replicator Dynamics Philippe Uyttendaele (joint work with Mandy Tak, Frank Thuijsman, Ronald Westra)

2. Global Replicator Dynamics

3. Local Replicator Dynamics • Field represent a torus

4. Local Replicator Dynamics • Random starting field

5. Local Replicator Dynamics • Focus on a cell

6. Local Replicator Dynamics • Interaction with each neighbors

7. Local Replicator Dynamics • Interaction with each neighbors

8. Local Replicator Dynamics • Interaction with each neighbors

9. Local Replicator Dynamics • Total fitness in a cell 12

10. Local Replicator Dynamics • Same procedure for entire field 54 12 12 12 30 12 36

11. Local Replicator Dynamics • Next generation

12. Global Replicator Dynamics

13. Local Replicator Dynamics • The GRD predicts to take over • Is there a possibility for this not to happen?

14. Local Replicator Dynamics • Stable pattern 12 18 18 12 18 18 12 24 12 24 24 12 12 12 12 18 24 24 24 18 12 12 18 18 12 12 18

15. In LRD all survive, in GRD not • This is a rare event in random simulations • Especially weak if mutations allowed

16. LRD • Are there other possible stable structures? • Can we find an easy Toy example like the Prisoner’s Dilemma?

17. In GRD all survive, in LRD not • Asymptotically stable for the GRD: (0.75 , 0.25) • Asymptotically stable for the LRD: (1 , 0)

18. LRD – Change in time/space

19. Fitness depends on availability of local resources Looks like predator prey models Populations moving around LRD – Resource Model

20. Multiple Populations Multiple Fields R1 R2 1, 2 Y1 Y2 0, 0 2, 1

21. Aligned Interactions x x x x Directional Patterns

22. What to do next? • Go deeper in the analysis of each scenario • Adapt the model based on the current one • Have a better understanding • What leads to “stable” situations? • Can we define stability? • What are the key features in the matrices?

23. Questions ? Beware, the snails are taking over the population