local replicator dynamics

Local Replicator Dynamics

Philippe Uyttendaele(joint work with Mandy Tak, Frank Thuijsman, Ronald Westra)

Global Replicator Dynamics

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Local Replicator Dynamics

• Field represent a torus

Local Replicator Dynamics

• Random starting field

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Local Replicator Dynamics

• Focus on a cell

Local Replicator Dynamics

• Interaction with each neighbors

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Local Replicator Dynamics

• Interaction with each neighbors

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Local Replicator Dynamics

• Interaction with each neighbors

Local Replicator Dynamics

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• Total fitness in a cell

Local Replicator Dynamics

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36Total 36 48 84

• Same procedure for entire field

Local Replicator Dynamics

• Next generation

Global Replicator Dynamics

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Local Replicator Dynamics

• The GRD predicts to take over

• Is there a possibility for this not to happen?

Local Replicator Dynamics

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• Stable pattern

In LRD all survive, in GRD not

• This is a rare event in random simulations

• Especially weak if mutations allowed

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LRD

• Are there other possible stable structures?

• Can we find an easy Toy example like the Prisoner’s Dilemma?

In GRD all survive, in LRD not

• Asymptotically stable for the GRD:

(0.75 , 0.25)

• Asymptotically stable for the LRD:

(1 , 0)

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LRD – Change in time/space

00)2cos(2

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)2cos(200

k

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

Multiple Populations Multiple Fields

1, 2

0, 0 2, 1

R1

Y2Y1

R2

Directional Patterns

• Aligned Interactions

x

x

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x

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?

Questions ?

Beware, the snails are taking over the population