The changing trends of <i>f</i>_<i>i</i>, <i>i</i> ∈ {1, 2, 3, 4} with <i>v</i> for the Moran process (first row) and the Wright-Fisher process (second row) when <i>u</i> is low.

<p>For all <i>r</i>, <i>f</i>₁ (first column) and <i>f</i>₂ (second column) have decreasing trends with <i>v</i>. For all <i>r</i>, the curve of <i>f</i>₃ (third column) with <i>v</i> is inverted U-shaped. For all <i>r</i>, The curve of <i>f</i>₄ with <i>v</i> (fourth column) has an increasing trend with a small perturbation near <i>v</i> = 0.01. When <i>r</i> = 1, the curve of <i>f</i>_<i>i</i>, <i>i</i> ∈ {1, 2, 3, 4} with <i>v</i> has a small perturbation near <i>v</i> = 1 for the Wright-Fisher process, but not for the Moran process. Parameters: <i>N</i> = 80, <i>M</i> = 7, <i>u</i> = 0.01, and <i>p</i> = 0.01.</p

<i>f</i>₁, <i>f</i>₂, <i>f</i>₃, and <i>f</i>₄.

<p><i>f</i>₁, <i>f</i>₂, <i>f</i>₃, and <i>f</i>₄.</p

The changing trends of <i>f</i>_<i>i</i>, <i>i</i> ∈ {1, 2, 3, 4} with <i>v</i> for the Moran process (first row) and the Wright-Fisher process (second row) when <i>u</i> is high.

<p>For all <i>r</i>, <i>f</i>₁ (first column) and <i>f</i>₂ (second column) decrease with <i>v</i>. For all <i>r</i>, <i>f</i>₃ (third column) and <i>f</i>₄ (fourth column) increases with <i>v</i>. Parameters: <i>N</i> = 80, <i>M</i> = 7, <i>u</i> = 1, and <i>p</i> = 0.01.</p

The changing trends of <i>f</i>_<i>i</i>, <i>i</i> ∈ {1, 2, 3, 4} with <i>u</i>.

<p>A, C: For the Moran process with <i>v</i> = 0 and the Wright-Fisher process with <i>v</i> = 0, <i>f</i>₁ and <i>f</i>₂ exhibit inverted U-shaped curves with <i>u</i>, but <i>f</i>₃ and <i>f</i>₄ exhibit U-shaped curves with <i>u</i>. B, D: For the Moran process with <i>v</i> = 0.05 and the Wright-Fisher process with <i>v</i> = 0.05, <i>f</i>₁, <i>f</i>₂, and <i>f</i>₃ exhibit inverted U-shaped curves with <i>u</i>, but <i>f</i>₄ exhibits a U-shaped curve with <i>u</i>. Parameters: <i>N</i> = 80, <i>M</i> = 7, <i>r</i> = 1, and <i>p</i> = 0.01.</p

The payoff matrix for the simplified version of the UG.

<p>The payoff matrix for the simplified version of the UG.</p

Comparison of theoretical results and simulated results in the Moran process (first row) and the Wright-Fisher process (second row).

<p>For the frequencies of selfishness (first column), fairness (second column), altruism (third column), and spite (fourth column), the difference of the theoretical values and the simulated values (averaged over 5 × 10⁸ − 10⁶ generations) is negligible when the selection intensity <i>δ</i> is sufficiently small, and is no longer negligible for other <i>δ</i>. Parameters: <i>N</i> = 50, <i>M</i> = 7, <i>u</i> = 0.1, <i>v</i> = 0.1, <i>r</i> = 1, and <i>p</i> = 0.01.</p

Γ₁, Γ₂ and Γ₃ for the Moran process (, , and respectively), and Γ₁, Γ₂ and Γ₃ for the Wright-Fisher process (, , and respectively), where Φ_<i>i</i>(<i>f</i>(<i>x</i>)), Ψ_<i>i</i>(<i>f</i>(<i>x</i>)), , and are abbreviated as Φ_<i>i</i>, , and .

<p>Γ₁, Γ₂ and Γ₃ for the Moran process (, , and respectively), and Γ₁, Γ₂ and Γ₃ for the Wright-Fisher process (, , and respectively), where Φ_<i>i</i>(<i>f</i>(<i>x</i>)), Ψ_<i>i</i>(<i>f</i>(<i>x</i>)), , and are abbreviated as Φ_<i>i</i>, , and .</p

Model schematic.

<p>(A) The simplified version of the UG with seven representative discrete strategies. (B) Migration of the range <i>r</i> when seven groups are located in a circle.</p

Evolutionary dynamics in structured populations satisfying the Tarnita-<i>σ</i> condition.

<p>A: The competition of selfishness (<i>S</i>₁) and fairness (<i>S</i>₂ and <i>S</i>₇) is influenced by altruism (<i>S</i>₃ and <i>S</i>₅) and spite (<i>S</i>₄ and <i>S</i>₆). B: For the seven-strategy competition, the scores of <i>S</i>₁, <i>S</i>₂, <i>S</i>₃, <i>S</i>₄, <i>S</i>₅, <i>S</i>₆, and <i>S</i>₇ are calculated.</p

For different <i>N</i>, comparison of <i>f</i>₁, <i>f</i>₂, <i>f</i>₃, and <i>f</i>₄ over the area spanned by <i>v</i> and <i>u</i>.

<p>For the Moran process (first row) and the Wright-Fisher process (second row), there are two phenomena <i>f</i>₂ > <i>f</i>₁ > <i>f</i>₃ > <i>f</i>₄ and <i>f</i>₂ > <i>f</i>₁ > <i>f</i>₄ > <i>f</i>₃ in the small population of <i>N</i> = 20; there only remains one phenomenon <i>f</i>₂ > <i>f</i>₁ > <i>f</i>₃ > <i>f</i>₄ in the moderate population of <i>N</i> = 80; <i>f</i>₂ > <i>f</i>₁ > <i>f</i>₃ > <i>f</i>₄ remains and there appears a new phenomenon <i>f</i>₂ > <i>f</i>₃ > <i>f</i>₁ > <i>f</i>₄ in the large population of <i>N</i> = 660. Parameters: <i>M</i> = 7, <i>r</i> = 1, and <i>p</i> = 0.01.</p