Replicator dynamics analysis of representative S-MIGs on 2-dimensional simplex.

Abstract

<p>The triangle represents the state space, Δ = {(<i>x</i>, <i>y</i>, <i>z</i>)*** : <i>x</i>, <i>y</i>, <i>z</i> ≥ 0, <i>x</i>+<i>y</i>+<i>z</i> = 1}, where <i>x</i>, <i>y</i>, and <i>z</i> are respectively the frequencies of the cooperative incentive-providers, cooperative incentive-non-providers, and non-cooperative incentive-non-providers. </p><p></p><p><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo></p><p><mn>1</mn><mn>2</mn></p><mo stretchy="false">)</mo><p></p><p></p>. (A) PR+R, (B) PP, (C) PB+RB(Full), and (D) RB. The abbreviations are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.t001" target="_blank">Table 1</a>. In (A), (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0) is unstable, so cooperation is never achieved regardless of the values of (<i>μ</i>, <i>δ</i>). In (B), the whole line <i>z</i> = 0 consists of fixed points, and thus, neutral drift is possible. In (C) and (D), (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0) is a locally asymptotically stable point depending on the values of (<i>μ</i>, <i>δ</i>), and thus, a cooperative regime can emerge. In (C), the unstable equilibrium in the internal part on <i>z</i> = 0, <i>K</i>_<i>z</i>, is a saddle, and that on <i>y</i> = 0, <i>K</i>_<i>y</i>, is a source. In (D), <i>K</i>_<i>z</i> is a source, while <i>K</i>_<i>y</i> is a saddle.<p></p