Individual-based simulation for moderately growing resource.

<p>Three representative phase portraits in <i>x</i> − <i>y</i>/<i>R</i>_<i>M</i> plane using different initial conditions when <i>e</i>_<i>c</i> < <i>r</i> < <i>e</i>_<i>d</i>. Panels show results obtained at powerful (left), moderate (middle), and weak (right) external institutions. Depending on the effectiveness of inspection and punishment a sustainable state can be reached for the first two cases. The specific values are <i>p</i> = 0.5 and <i>β</i> = 0.5 in panel A; <i>p</i> = 0.05 and <i>β</i> = 0.5 in panel B; and <i>p</i> = 0.01 and <i>β</i> = 0.1 in panel C. Other parameters are <i>r</i> = 0.6, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>α</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5 for all cases.</p

Replicator dynamics for rapidly growing resource.

<p>Top panels show the time evolution of the fraction of cooperators and the resource abundance ratio for different parameter values when <i>e</i>_<i>c</i> < <i>e</i>_<i>d</i> < <i>r</i>. Bottom panels show the related phase portraits on <i>x</i> − <i>y</i>/<i>R</i>_<i>m</i> plane. Parameters for Panels A and B: <i>r</i> = 1.0, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>p</i> = 0.5, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5; for Panels C and D: <i>r</i> = 1.0, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>p</i> = 0.2, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5; for Panels E and F: <i>r</i> = 1.0, <i>N</i> = 100, <i>R</i>_<i>m</i> = 100, <i>p</i> = 0.1, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5. Due to the large intrinsic growth rate, the environmental resource will never be depleted. The strength of external institutions determines the relation of competing strategies in the equilibrium state.</p

A representative plot of evolutionary outcomes on the phase plane.

<p>Different colors are used to distinguish qualitatively different solutions in the parameter space (<i>r</i>, <i>pβ</i>). This plot highlights that the inner dynamical feature of renewable resource could be a decisive factor that can derogate the expected consequence of punishment.</p

Individual-based simulation for rapidly growing resource.

<p>Three representative phase portraits in <i>x</i> − <i>y</i>/<i>R</i>_<i>M</i> plane using different initial conditions when <i>e</i>_<i>c</i> < <i>e</i>_<i>d</i> < <i>r</i>. Panels show results obtained at powerful (left), moderate (middle), and weak (right) external institutions. Due to large growth rate a sustainable state can always be maintained. The strength of inspection and punishment determines only the level of this state. The specific values are <i>p</i> = 0.5, 0.2, and 0.1 for panel A, B, and C, respectively. Other parameters are <i>r</i> = 1.0, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5 for all cases.</p

Replicator dynamics for moderately growing resource.

<p>Top panels show the time evolution of the fraction of cooperators and the resource abundance ratio for different parameter values when <i>e</i>_<i>c</i> < <i>r</i> < <i>e</i>_<i>d</i>. Bottom panels show the related phase portraits on <i>x</i> − <i>y</i>/<i>R</i>_<i>m</i> plane. Parameters for Panels A and B: <i>r</i> = 0.6, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>p</i> = 0.5, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5; for Panels C and D: <i>r</i> = 0.6, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>p</i> = 0.05, <i>α</i> = 0.5, <i>β</i> = 0.5, and <i>b</i>_<i>m</i> = 0.5; for Panels E and F: <i>r</i> = 0.6, <i>N</i> = 1000, <i>R</i>_<i>m</i> = 1000, <i>p</i> = 0.01, <i>α</i> = 0.5, <i>β</i> = 0.1, and <i>b</i>_<i>m</i> = 0.5. These plots suggest that a sustainable state can be reached for appropriate inspection and punishment level, but the depleted environment state cannot be avoided if these institutions are ineffective.</p