Mutualisms: cooperation between species: Difference between revisions
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[[Image:Mutualism - phase transition r1.png|thumb|300px|The scaling of '''a''' \(\bar \rho_a\) and '''b''' \(\chi_a\) versus \((r_1-r)/r_1\) follows the directed percolation universality class (solid lines indicate theoretical expectations).]] | [[Image:Mutualism - phase transition r1.png|thumb|300px|The scaling of '''a''' \(\bar \rho_a\) and '''b''' \(\chi_a\) versus \((r_1-r)/r_1\) follows the directed percolation universality class (solid lines indicate theoretical expectations).]] | ||
Near the extinction threshold, \(r_1\), isolated clusters of cooperators perform a branching and annihilating random walk with almost perfect correlations between lattices. The characteristic spatial patterns are captured by snapshots of the configuration in each lattice as well as the pair distributions (see [[#Symmetric cooperation: random walk| Symmetric cooperation: random walk]]). | Near the extinction threshold, \(r_1\), isolated clusters of cooperators perform a branching and annihilating random walk with almost perfect correlations between lattices. The characteristic spatial patterns are captured by snapshots of the configuration in each lattice as well as the pair distributions (see [[#Symmetric cooperation: random walk|Symmetric cooperation: random walk]]). | ||
Coordination between lattices is almost perfect with very few \(CD\) and \(DC\) pairs. This is of the essence for the survival of cooperation. \(CC\) interactions are the bulwark against, or at least help to compensate for exploitation by defectors. Interestingly, the emerging spatial configurations are very similar to those in the intra-species donation game on a single lattice. | Coordination between lattices is almost perfect with very few \(CD\) and \(DC\) pairs. This is of the essence for the survival of cooperation. \(CC\) interactions are the bulwark against, or at least help to compensate for exploitation by defectors. Interestingly, the emerging spatial configurations are very similar to those in the intra-species donation game on a single lattice. | ||
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In agreement with theoretical expectations the critical transition at \(r_1\) clearly belongs to the two-dimensional directed percolation universality class: in both layers the frequency of cooperation scales with \(\bar\rho_a \propto (r_{1}-r)^\beta\) for \(\beta=0.580(3)\) and their fluctuations scale with \(\chi_a \propto (r_{1}-r)^{-\gamma}\) for \(\gamma=0.35(1)\) when \(r\to r_1\) from below (as shown on the right) | In agreement with theoretical expectations the critical transition at \(r_1\) clearly belongs to the two-dimensional directed percolation universality class: in both layers the frequency of cooperation scales with \(\bar\rho_a \propto (r_{1}-r)^\beta\) for \(\beta=0.580(3)\) and their fluctuations scale with \(\chi_a \propto (r_{1}-r)^{-\gamma}\) for \(\gamma=0.35(1)\) when \(r\to r_1\) from below (as shown on the right) | ||
When lowering the cost-to-benefit ratio, \(r_2<r<r_1\), the frequency of cooperation gradually increases with decreasing \(r\), while the coordination between layers decreases. For \(r\) close to \(r_2\) the frequency of all four strategy pairs are essentially the same, which supports that configurations are largely uncorrelated. | When lowering the cost-to-benefit ratio, \(r_2<r<r_1\), the frequency of cooperation gradually increases with decreasing \(r\), while the coordination between layers decreases. For \(r\) close to \(r_2\) the frequency of all four strategy pairs are essentially the same, which supports that configurations are largely uncorrelated (see [[#Symmetric cooperation: uncorrelated|Symmetric cooperation: uncorrelated]]). | ||
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[[Image:Mutualism - phase transition r2.png|thumb|300px|The scaling of the order parameter \(\Phi=\vert\bar\rho_{+1}-\bar\rho_{-1}\vert\) as a function of \((r_2-r)/r_2\) is in line with the universality class of spontaneous magnetization in the Ising model. '''b''' The scaling of the fluctuations of \(\Phi\) (\(\bullet\) and \(\circ\)) as well as \(\chi_a\) (\(\Box\) and \(\times\)) above and below the critical threshold \(r_2\). The \(\Phi\) data reproduces the theoretical expectations (solid lines with slopes \(1/8\) in '''a''' and \(-7/4\) in '''b''') within statistical errors (omitted for clarity).]] | [[Image:Mutualism - phase transition r2.png|thumb|300px|The scaling of the order parameter \(\Phi=\vert\bar\rho_{+1}-\bar\rho_{-1}\vert\) as a function of \((r_2-r)/r_2\) is in line with the universality class of spontaneous magnetization in the Ising model. '''b''' The scaling of the fluctuations of \(\Phi\) (\(\bullet\) and \(\circ\)) as well as \(\chi_a\) (\(\Box\) and \(\times\)) above and below the critical threshold \(r_2\). The \(\Phi\) data reproduces the theoretical expectations (solid lines with slopes \(1/8\) in '''a''' and \(-7/4\) in '''b''') within statistical errors (omitted for clarity).]] | ||
Intriguingly, lowering \(r\) further to \(r_3<r<r_2\) with \(r_3\approx0.00041(3)\) results in spontaneous symmetry breaking with different frequencies in each layer. | Intriguingly, lowering \(r\) further to \(r_3<r<r_2\) with \(r_3\approx0.00041(3)\) results in spontaneous symmetry breaking with different frequencies in each layer (see [[#Asymmetric cooperation|Asymmetric cooperation]]). | ||
The distribution of cooperators and defectors in each layer is almost complementary. Clusters, or regions of cooperators, in one layer are matched by defectors in the other. As a consequence, the lattices consist mostly of \(CD\) and \(DC\) pairs. Because of the distinctly different frequencies of cooperation in each lattice, either \(CD\) or \(DC\) pairs dominate. However, note that which pair dominates is of no consequence, because species interactions are symmetric, and hence it is merely a consequence of which species is labelled \(X\) and which \(Y\). Defectors in one species manage to exploit and take advantage of cooperators in the other species. However, exploitation is ephemeral and the scale is limited both spatially as well as temporarily. | The distribution of cooperators and defectors in each layer is almost complementary. Clusters, or regions of cooperators, in one layer are matched by defectors in the other. As a consequence, the lattices consist mostly of \(CD\) and \(DC\) pairs. Because of the distinctly different frequencies of cooperation in each lattice, either \(CD\) or \(DC\) pairs dominate. However, note that which pair dominates is of no consequence, because species interactions are symmetric, and hence it is merely a consequence of which species is labelled \(X\) and which \(Y\). Defectors in one species manage to exploit and take advantage of cooperators in the other species. However, exploitation is ephemeral and the scale is limited both spatially as well as temporarily. | ||
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==== Bursts of defection ==== | ==== Bursts of defection ==== | ||
[[Image:Mutualism - phase transition r3.png|thumb|150px|<div style="width:296px">Scaling of \(\chi_a\) in each layer (\(\Box\) and \(\times\)) versus \((r-r_3)/r_3\), which is driven by bursts of defection. For reference the solid line shows a power law with exponent \(-1.33\).</div>]] | [[Image:Mutualism - phase transition r3.png|thumb|150px|<div style="width:296px">Scaling of \(\chi_a\) in each layer (\(\Box\) and \(\times\)) versus \((r-r_3)/r_3\), which is driven by bursts of defection. For reference the solid line shows a power law with exponent \(-1.33\).</div>]] | ||
Finally, for \(0<r<r_3\), the population relaxes into one of its four absorbing states depending on the initial configuration. In fact, the most striking dynamical phenomenon of this model manifests itself in the diverging fluctuations of \(C\) frequencies, \(\chi_a\), for \(r\to r_3\) (see [[#Bursts of defection|Bursts of defection]]). | |||
In particular, no theoretical arguments exist to date that explain and justify the observed power law behaviour of \(\chi_a \propto (r-r_3)^{-\gamma}\) for \(r\to r_3\) from above with \(\gamma=1.33(5)\). The divergence of \(\chi_a\) is related to the emergence, growth and elimination of homogeneous \(DD\) domains. Spikes in \(DD\) frequencies drive the dynamics of bursts. | |||
This type of bursts are driven by sufficiently large spatial domains of an unstable phase. Evidently, their extension and frequency depends on \(r\) and other parameters. In particular, bursts become larger and rarer when decreasing \(r\). Once a burst reaches sizes comparable to the entire lattice, it can drive the evolution towards absorbing states. Interestingly, the absorbing \(DD\) state is rarely observed because it is prone to invasion by any remaining \(CC\) islands, which then pave the way for \(CD\) or \(DC\) phases to grow and take over. This is in stark contrast to intra-species models where cooperation continuously increases for decreasing \(r\), that is as the social dilemma becomes more benign. | |||
Another peculiarity of the burst dynamics is the feedback between their creation and extinction and the spatial configuration of the asymmetric phase: for \(r\) close to \(r_3\), the asymmetry of cooperation, \(\Phi = |\bar\rho_{+1} - \bar\rho_{-1}|\), decreases with \(r\) while the burst sizes increase. | |||
For \(r<r_3\) the population tends to get absorbed in the fully asymmetric \(CD\) or \(DC\) states. Once one layer is homogeneous the dynamics in the other layer is dominated by neutral drift because the difference in payoffs between cooperators and defectors merely amounts to \(4r\). Hence, for small \(r\), the probability to update the strategy essentially reduces to a coin toss with a slight bias in favour of defectors. The probability to fixate in one or the other absorbing state is thus essentially given by the frequency of strategies in the heterogeneous layer at the time the other layer turned homogeneous. For example, if one layer becomes homogeneous in \(C\), then the frequency of \(C\)’s in the other layer is likely \(<1/2\), due to the asymmetry. Thus, the \(CD\) absorbing state is more likely than \(CC\). Conversely, if one layer becomes homogeneous in $D$, then the frequency of \(C\)’s in the other layer is likely \(>1/2\) and hence the \(DC\) absorbing state is more likely than \(DD\). Finally, it is more likely that one layer becomes homogeneous in \(D\) because isolated \(C\)’s (or small \(C\) clusters) are more easily eliminated than \(D\)’s. Thus, even though all four absorbing states may be reached, in principle, the probabilities to do so are very different and most likely the populations approach heterogeneous absorbing states, whereas homogeneous cooperation across lattices is a highly unlikely outcome. | |||
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