Evolutionary Games and Population Dynamics / Selection-diffusion / Gallery

• Spatio-temporal chaos arising in ecological public goods. Parameters: N=8, r=2.34, c=1, b=1, d=1.2, DC=1, DD=2, L=400, dt=0.1, dx=2; initial configuration: empty space with centered disk with diameter L/5 and density of 0.1 of cooperators as well as defectors.
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  [math]\displaystyle{ t=0 }[/math]

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  [math]\displaystyle{ t=250 }[/math]

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  [math]\displaystyle{ t=500 }[/math]

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  [math]\displaystyle{ t=750 }[/math]

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  [math]\displaystyle{ t=1000 }[/math]

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  [math]\displaystyle{ t=1250 }[/math]

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  [math]\displaystyle{ t=1500 }[/math]

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  [math]\displaystyle{ t=1750 }[/math]

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  [math]\displaystyle{ t=2000 }[/math]

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  [math]\displaystyle{ t=2250 }[/math]

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  [math]\displaystyle{ t=2500 }[/math]

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  [math]\displaystyle{ t=2750 }[/math]

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  [math]\displaystyle{ t=3000 }[/math]

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  [math]\displaystyle{ t=3250 }[/math]

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  [math]\displaystyle{ t=3500 }[/math]

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  [math]\displaystyle{ t=3750 }[/math]

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  [math]\displaystyle{ t=4000 }[/math]

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  [math]\displaystyle{ t=4250 }[/math]

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  [math]\displaystyle{ t=4500 }[/math]

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  [math]\displaystyle{ t=4750 }[/math]

Brave New World of Dynamical Pattern Formation Typical patterns for different efficiency of the public good [math]\displaystyle{ r }[/math] and ratios of the diffusion coefficients. For [math]\displaystyle{ r \gt r_\text{Hopf} }[/math] the co-existence equilibrium is stable and unstable otherwise, i.e. the population would invariably disappear in the absence of spatial extension. The spatial dynamics can be categorized into different kinds of pattern formation processes: homogenous distributions (yellow frame), diffusion induced instability - Turing patterns (green), diffusion induced co-existence (red), spatio-temporal chaos (blue) and extinction (black).

Effects of initial configurations Same as above but for symmetrical initial configurations with a disk in the center with equal densities of cooperators and defectors. The initial symmetry is often largely preserved by the deterministic dynamics but more importantly, the characteristic features of the emerging patterns remain unchanged.