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Evolutionary Games and Population Dynamics/Selection-diffusion: Difference between revisions
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Evolutionary Games and Population Dynamics/Selection-diffusion (view source)
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== Modes of pattern formation == | == Modes of pattern formation == | ||
The rich dynamics of Ecological Public Goods in spatial settings can be explored by interactive simulations or visualized through movies of high accuracy simulations. Clicking on the images loads interactive real-time | The rich dynamics of Ecological Public Goods in spatial settings can be explored by interactive simulations or visualized through movies of high accuracy simulations. Clicking on the images loads interactive real-time ''EvoLudo'' simulations that illustrate the characteristics of the corresponding dynamical regime. Because the simulations require significant computational power, high resolution movies are provided as an alternative (requires Quicktime 7 or higher - movies are H.264 encoded). The densities of cooperator and defectors across space are indicated by the brightness of the green and red color components, respectively. Thus, regions of co-existence appear yellow and black regions are vacant. The initial configuration is a disk of homogeneous cooperator and defector densities centered in an empty plane (no-flux boundaries). | ||
{{Legend:Gradient|label=Cooperator density|color=green|min=Low|max=High||label1=Defector density|color1=red|label2=Population density|color2=yellow}} | {{Legend:Gradient|label=Cooperator density|color=green|min=Low|max=High||label1=Defector density|color1=red|label2=Population density|color2=yellow}} | ||
{{Legend:Gradient}} | {{Legend:Gradient}} | ||
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<div class="lab_description EcoPGG"> | <div class="lab_description EcoPGG"> | ||
[[Image:Spatial Ecological PGG - Chaos.jpg|left|200px]] | [[Image:Spatial Ecological PGG - Chaos.jpg|left|200px]] | ||
==== [[ | ==== [[EvoLudoLab: Spatial Ecological PGG - Chaos|Spatio-temporal chaos]]==== | ||
Ecological Public Goods Games in spatial settings can exhibit chaotic dynamics and produce fascinating ever changing patterns. This movie illustrates the onset of chaos when starting from a symmetrical initial configuration. The deterministic dynamics should, in principle, preserve the symmetry but instead the symmetry is maintained only for some time and then breaks down due to limitations of the numerical integration. The exponential amplification of arbitrarily small disturbances is the hallmark chaotic systems. In the absence of spatial dimensions the population would be unable to persist. | Ecological Public Goods Games in spatial settings can exhibit chaotic dynamics and produce fascinating ever changing patterns. This movie illustrates the onset of chaos when starting from a symmetrical initial configuration. The deterministic dynamics should, in principle, preserve the symmetry but instead the symmetry is maintained only for some time and then breaks down due to limitations of the numerical integration. The exponential amplification of arbitrarily small disturbances is the hallmark chaotic systems. In the absence of spatial dimensions the population would be unable to persist. | ||
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<div class="lab_description EcoPGG"> | <div class="lab_description EcoPGG"> | ||
[[Image:Spatial Ecological PGG - Intermittency.jpg|left|200px]] | [[Image:Spatial Ecological PGG - Intermittency.jpg|left|200px]] | ||
==== [[ | ==== [[EvoLudoLab: Spatial Ecological PGG - Intermittency |Intermittent activity]]==== | ||
Between the regimes of chaotic dynamics and static patterns an interesting region of intermittent activity occurs: The formation of quasi-static patterns alternates with rapid changes leading to global rearrangement and redistribution of cooperators and defectors. This movie illustrates the successive periods of stasis with intermittent bursts of activity. At first the bursts are synchronized across space but this breaks down over time generating states where large parts are largely static but get regularly stirred up by a wave of change. | Between the regimes of chaotic dynamics and static patterns an interesting region of intermittent activity occurs: The formation of quasi-static patterns alternates with rapid changes leading to global rearrangement and redistribution of cooperators and defectors. This movie illustrates the successive periods of stasis with intermittent bursts of activity. At first the bursts are synchronized across space but this breaks down over time generating states where large parts are largely static but get regularly stirred up by a wave of change. | ||
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<div class="lab_description EcoPGG"> | <div class="lab_description EcoPGG"> | ||
[[Image:Spatial Ecological PGG - diffusion induced instability.jpg|left|200px]] | [[Image:Spatial Ecological PGG - diffusion induced instability.jpg|left|200px]] | ||
==== [[ | ==== [[EvoLudoLab: Spatial Ecological PGG - Diffusion induced instability|Diffusion induced instability - Turing patterns]]==== | ||
In the absence of space, cooperators and defectors can co-exist in a stable equilibrium Q. In spatial settings, the corresponding homogeneous strategy distribution is often unstable. Diffusion (or migration) generates an activator-inhibitor system that leads to spontaneous pattern formation also known as Turing patterns. Any deviation from Q is amplified by cooperators (activators) but suppressed by defectors (inhibitors). These antagonistic forces give rise to the formation of complex patterns. Any small local disturbance propagates through the system and induces stable heterogeneous strategy distributions. Further local disturbances may give rise to rearrangements of the patterns but then quickly relax into another qualitatively indistinguishable distribution of cooperators and defectors. In the movie, the inhomogeneity of the initial configuration triggers the formation of Turing patterns. | In the absence of space, cooperators and defectors can co-exist in a stable equilibrium Q. In spatial settings, the corresponding homogeneous strategy distribution is often unstable. Diffusion (or migration) generates an activator-inhibitor system that leads to spontaneous pattern formation also known as Turing patterns. Any deviation from Q is amplified by cooperators (activators) but suppressed by defectors (inhibitors). These antagonistic forces give rise to the formation of complex patterns. Any small local disturbance propagates through the system and induces stable heterogeneous strategy distributions. Further local disturbances may give rise to rearrangements of the patterns but then quickly relax into another qualitatively indistinguishable distribution of cooperators and defectors. In the movie, the inhomogeneity of the initial configuration triggers the formation of Turing patterns. | ||
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<div class="lab_description EcoPGG"> | <div class="lab_description EcoPGG"> | ||
[[Image:Spatial Ecological PGG - diffusion induced coexistence.jpg|left|200px]] | [[Image:Spatial Ecological PGG - diffusion induced coexistence.jpg|left|200px]] | ||
==== [[ | ==== [[EvoLudoLab: Spatial Ecological PGG - Diffusion induced coexistence |Diffusion induced coexistence]]==== | ||
In the absence of space, the co-existence equilibrium Q is unstable such that cooperator and defector frequencies exhibit oscillations with increasing amplitudes until eventually the population goes extinct. In spatial settings with diffusion, cooperators and defectors again form an activator-inhibitor system but this time emerging spatial patterns are responsible for the survival of the population and permit stable co-existence of cooperators and defectors. This movie illustrates the emergence of spatial patterns through diffusion induced co-existence. The setup is the same as for the Turing patterns above, only the yield of the public good is lower. | In the absence of space, the co-existence equilibrium Q is unstable such that cooperator and defector frequencies exhibit oscillations with increasing amplitudes until eventually the population goes extinct. In spatial settings with diffusion, cooperators and defectors again form an activator-inhibitor system but this time emerging spatial patterns are responsible for the survival of the population and permit stable co-existence of cooperators and defectors. This movie illustrates the emergence of spatial patterns through diffusion induced co-existence. The setup is the same as for the Turing patterns above, only the yield of the public good is lower. | ||
