Local Replicator Dynamics
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Transcript of Local Replicator Dynamics
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Local Replicator Dynamics
Philippe Uyttendaele(joint work with Mandy Tak, Frank Thuijsman, Ronald Westra)
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Global Replicator Dynamics
1713
0012
060
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Local Replicator Dynamics
• Field represent a torus
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Local Replicator Dynamics
• Random starting field
1713
0012
060
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1713
0012
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Local Replicator Dynamics
• Focus on a cell
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Local Replicator Dynamics
• Interaction with each neighbors
1713
0012
060
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Local Replicator Dynamics
• Interaction with each neighbors
1713
0012
060
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1713
0012
060
Local Replicator Dynamics
• Interaction with each neighbors
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Local Replicator Dynamics
12
1713
0012
060
• Total fitness in a cell
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Local Replicator Dynamics
12
12
12
12
30
54
36Total 36 48 84
• Same procedure for entire field
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Local Replicator Dynamics
• Next generation
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Global Replicator Dynamics
1713
0012
060
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Local Replicator Dynamics
• The GRD predicts to take over
• Is there a possibility for this not to happen?
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Local Replicator Dynamics
2424
24
12
12
12
12
12
1224
24 2418
18 18
18
18 18
18 18
18
12 12
12
1212
12
• Stable pattern
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In LRD all survive, in GRD not
• This is a rare event in random simulations
• Especially weak if mutations allowed
1713
0012
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LRD
• Are there other possible stable structures?
• Can we find an easy Toy example like the Prisoner’s Dilemma?
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In GRD all survive, in LRD not
• Asymptotically stable for the GRD:
(0.75 , 0.25)
• Asymptotically stable for the LRD:
(1 , 0)
01
30
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LRD – Change in time/space
00)2cos(2
0
)2cos(200
k
xt
k
xt
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LRD – Resource Model• Fitness depends on availability of local resources• Looks like predator prey models• Populations moving around
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Multiple Populations Multiple Fields
1, 2
0, 0 2, 1
R1
Y2Y1
R2
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Directional Patterns
• Aligned Interactions
x
x
x
x
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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?
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Questions ?
Beware, the snails are taking over the population