Supplemental Material for: Judgment on the evolution state of asymmetric games

The file includes the derivation process of formulas, proofs, figures, etc

Supplementary Information from Stability analysis of evolutionary dynamics of 2 × 2 × 2 asymmetric games

The ESM file provides proofs supporting the conclusions in the main text, examples, and code

(color online). The relationships of five species in the Jungle game.

<p>Arrows point from predator to prey. <i>S</i>₁ and <i>S</i>₂ can prey three species and be hunt by one species; <i>S</i>₃ can prey two species and be hunt by two species; <i>S</i>₄ and <i>S</i>₅ can prey one species and be hunt by three species.</p

(color online) Coexistence of species in example 1 using Monte Carlo simulation. <i>L</i> = 400.

<p>All five species coexist in the red region. Species <i>S</i>₁<i>S</i>₄<i>S</i>₅ coexist in the orange region. Species <i>S</i>₁<i>S</i>₂<i>S</i>₅ coexist in the light yellow region. Species <i>S</i>₁<i>S</i>₃<i>S</i>₅ coexist in the deep yellow region. Only <i>S</i>₅ remains in the green region. Only <i>S</i>₁ or <i>S</i>₂ remains in the blue region.</p

(color online) Densities of five species in Monte Carlo simulation. <i>L</i> = 200,<i>k</i>_{<i>i</i>, <i>j</i>} = 1.

<p>After about 500 time steps, the species <i>S</i>₄ and <i>S</i>₃ extinct and species <i>S</i>₁, <i>S</i>₂ and <i>S</i>₅ coexist.</p

The area of region II in Fig 3 with different <i>p</i>₁.

<p>There exists <i>P</i> ≈ 1.149 making the smallest area at <i>p</i>₁ = <i>P</i>. When <i>p</i>₁ < <i>P</i>, the area decreases with the increasing <i>p</i>₁; when <i>p</i>₁ > <i>P</i>, the area increases with the increasing <i>p</i>₁.</p

Attachment of Supplementary Information from Stability analysis of evolutionary dynamics of 2 × 2 × 2 asymmetric games

This ESM file provides code and figures, especially dynamic figures

The biodiversity of the Jungle game in the second example.

<p>(a) <i>p</i>₁ ≤ 1; (b) ; (c) . Species <i>S</i>₁, <i>S</i>₂ and <i>S</i>₅ coexist in region I (green); all five species coexist in region II (white); and species <i>S</i>₁, <i>S</i>₄ and <i>S</i>₅ coexist in region III (yellow).</p

(color online) <i>p</i> = 0.5, <i>s</i> = 1.2.

<p>Densities of species under different population sizes. (a) <i>L</i> = 100. (b) <i>L</i> = 200. (c) <i>L</i> = 400. (d) <i>L</i> = 800. (a) Species <i>S</i>₂ and <i>S</i>₃ extinct after 10000 MCS, species <i>S</i>₁, <i>S</i>₄ and <i>S</i>₅ coexist. In (b), (c) and (d), all the five species coexist, the densities fluctuation decreases with the increasing <i>L</i>.</p

The biodiversity of the Jungle game in the first example.

<p>Species <i>S</i>₁, <i>S</i>₂ and <i>S</i>₅ coexist in region I; all five species coexist in region II; and species <i>S</i>₁, <i>S</i>₄ and <i>S</i>₅ coexist in region III.</p