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. 2011 Nov 18;6(11):e27523. doi: 10.1371/journal.pone.0027523

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Zhang, Hui.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

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Brown part and part iv, , have two boundary equilibriums: one for cooperators; the other for defectors. Yellow part, , has only one boundary equilibrium for cooperators. Cyan part, , has two boundary equilibriums both for cooperators. The area encircled by blue curves indicates the existence of interior equilibrium of cooperation-defect coexistence. The three red lines on (A) and (B), from bottom to top, indicate the node-focus bifurcation, Hopf bifurcation and the heteroclinic bifurcation, respectively, and the red lines on (C) and (D) indicate the node-focus bifurcation. Area on the left side of the vertical dotted line indicates the invasion condition for cooperation in the frequency-dependent selection (i.e. ). Parameters are and m = 0.1, 0.2, 0.3 and 0.55, respectively, for panel (A) to (D).

Inline graphic — Brown part and part iv, , have two boundary equilibriums: one for cooperators; the other for defectors. Yellow part, , has only one boundary equilibrium for cooperators. Cyan part, , has two boundary equilibriums both for cooperators. The area encircled by blue curves indicates the existence of interior equilibrium of cooperation-defect coexistence. The three red lines on (A) and (B), from bottom to top, indicate the node-focus bifurcation, Hopf bifurcation and the heteroclinic bifurcation, respectively, and the red lines on (C) and (D) indicate the node-focus bifurcation. Area on the left side of the vertical dotted line indicates the invasion condition for cooperation in the frequency-dependent selection (i.e. ). Parameters are and m = 0.1, 0.2, 0.3 and 0.55, respectively, for panel (A) to (D).