860
edits
Origin of Cooperators and Defectors: Difference between revisions
From EvoLudo
Origin of Cooperators and Defectors (view source)
Revision as of 17:52, 7 August 2009
, 7 August 2009→Branching point and repellor
No edit summary |
|||
| Line 62: | Line 62: | ||
[[Image:Continuous Snowdrift Game - Branching (sqrt).png|left|200px]] | [[Image:Continuous Snowdrift Game - Branching (sqrt).png|left|200px]] | ||
==== [[VirtualLab: Continuous Snowdrift Game - Branching (sqrt)|Branching point and repellor]] ==== | ==== [[VirtualLab: Continuous Snowdrift Game - Branching (sqrt)|Branching point and repellor]] ==== | ||
More complicated cost and benefit functions lead to very interesting dynamics but generally they make a complete analysis impossible. Here we set <math>B(x) = b \sqrt{(x+y)}</math> and <math>C(x) = \ln(c x+1)</math>. This may lead to the simultaneous occurrence of a branching point and a repellor. The results are summarized in the figure to the left. Starting to the left of the repellor (dash-dotted vertical line) investments decrease until they reach the branching point (dashed vertical line) and then two strategies co-exist and diverge until the upper branch reaches the maximum investment, i.e. hits the boundary of the trait range. Note that the dimorphic population after the branching point does no longer 'see' the repellor near <math>x = 4</math>. The small inset show another simulation run starting to the right of the repellor. The trait simply increases until it hits the boundary. | More complicated cost and benefit functions lead to very interesting dynamics but generally they make a complete analysis impossible. Here we set <math>B(x) = b \sqrt{(x+y)}+a</math> and <math>C(x) = \ln(c x+1)</math>. This may lead to the simultaneous occurrence of a branching point and a repellor. The results are summarized in the figure to the left. Starting to the left of the repellor (dash-dotted vertical line) investments decrease until they reach the branching point (dashed vertical line) and then two strategies co-exist and diverge until the upper branch reaches the maximum investment, i.e. hits the boundary of the trait range. Note that the dimorphic population after the branching point does no longer 'see' the repellor near <math>x = 4</math>. The small inset show another simulation run starting to the right of the repellor. The trait simply increases until it hits the boundary. | ||
<br style="clear:both" /> | <br style="clear:both" /> | ||
</div> | </div> | ||
| Line 68: | Line 68: | ||
<div class="lab_description CSD"> | <div class="lab_description CSD"> | ||
[[Image:Continuous Snowdrift Game - Branching (exp).png|left|200px]] | [[Image:Continuous Snowdrift Game - Branching (exp).png|left|200px]] | ||
==== [[VirtualLab: Continuous Snowdrift Game - Branching (exp)|Branching point and repellor - intermediate investment levels]] ==== | ==== [[VirtualLab: Continuous Snowdrift Game - Branching (exp)|Branching point and repellor - intermediate investment levels]] ==== | ||
So far, whenever branching occurred the upper branch, i.e. investments increased until they reached the boundary of the trait range. In order to illustrate that this must not be the case consider <math>B(x+y) = b (1-\exp(-x))</math> and <math>C(x) = \ln(c x+1)</math>. As shown in the figure to the left, it may now occur that initially high investment levels decrease until the branching point is reached and then the upper branch reaches saturation around <math>x = 2.1</math>. The fact that at low <math>x</math> there exists also a repellor becomes evident only when starting with very small <math>x_0</math>. In that case the trait quickly reaches zero and remains there as shown in the little inset. | So far, whenever branching occurred the upper branch, i.e. investments increased until they reached the boundary of the trait range. In order to illustrate that this must not be the case consider <math>B(x+y) = b (1-\exp(-x))</math> and <math>C(x) = \ln(c x+1)</math>. As shown in the figure to the left, it may now occur that initially high investment levels decrease until the branching point is reached and then the upper branch reaches saturation around <math>x = 2.1</math>. The fact that at low <math>x</math> there exists also a repellor becomes evident only when starting with very small <math>x_0</math>. In that case the trait quickly reaches zero and remains there as shown in the little inset. | ||
