Mutualisms: cooperation between species: Difference between revisions
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\begin{array}{cc} | \begin{array}{cc} | ||
\label{eq:dg} | |||
\tag{1} | \tag{1} | ||
& C\qquad D\\ | & C\qquad D\\ | ||
\begin{matrix} | \begin{matrix} | ||
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Thus, both players prefer mutual cooperation over mutual defection, yet defection yields the higher payoff regardless of what the partner does and hence the conflict of interest arises. | Thus, both players prefer mutual cooperation over mutual defection, yet defection yields the higher payoff regardless of what the partner does and hence the conflict of interest arises. | ||
Rescaling reduces the interaction in Eq. \eqref{eq:dg} to a single parameter: | |||
\begin{equation} | |||
\label{eq:dgr} | |||
\tag{2} | |||
{\bf A}= | |||
\begin{pmatrix} 1-r & -r\\ | |||
1 & 0 \end{pmatrix}, | |||
\end{equation} | |||
where \(r=c/b\) denotes the cost-to-benefit ratio with \(0\lt r\lt 1\). Moreover, for dynamics that only depend on payoff differences, a constant can be added to ensure that all payoffs are non-negative: | |||
\begin{equation} | |||
\label{eq:dgr+} | |||
\tag{3} | |||
{\bf A}= | |||
\begin{pmatrix} 1 & 0\\ | |||
1+r & r \end{pmatrix}. | |||
\end{equation} | |||
The problem of cooperation is exacerbated for actions that bestow benefits not just to other individuals but to those of another species. In particular it is crucial to clearly distinguish between the acts of cooperation of an individual and mutualistic interactions between individuals. Accomplishing mutually beneficial interactions requires coordination of cooperative acts between species. | The problem of cooperation is exacerbated for actions that bestow benefits not just to other individuals but to those of another species. In particular it is crucial to clearly distinguish between the acts of cooperation of an individual and mutualistic interactions between individuals. Accomplishing mutually beneficial interactions requires coordination of cooperative acts between species. | ||
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\begin{align} | \begin{align} | ||
\label{eq:re} | \label{eq:re} | ||
\tag{ | \tag{4} | ||
\dot x &= x(\xi_C - \bar\xi) = x(1-x)(\xi_C - \xi_D)\\ | \dot x &= x(\xi_C - \bar\xi) = x(1-x)(\xi_C - \xi_D)\\ | ||
\end{align} | \end{align} | ||
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\begin{align} | \begin{align} | ||
\label{eq:remut} | \label{eq:remut} | ||
\tag{ | \tag{5} | ||
\dot x &= x(\xi_C - \bar\xi)\\ | \dot x &= x(\xi_C - \bar\xi)\\ | ||
\notag | |||
\dot y &= y(\zeta_C - \bar\zeta), | \dot y &= y(\zeta_C - \bar\zeta), | ||
\end{align} | \end{align} | ||
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Eq. \eqref{eq:remut} reduces to \(\dot x = x(1-x)(-c_x) \lt 0\) and \(\dot y = y(1-y)(-c_y) \lt 0\) such that in both species cooperators dwindle and eventually go extinct. Interestingly, this happens regardless of the magnitude of foregone benefits, \(b_x,b_y\). Thus, unsurprisingly the demise of cooperation in inter-species interactions remains unchanged. | Eq. \eqref{eq:remut} reduces to \(\dot x = x(1-x)(-c_x) \lt 0\) and \(\dot y = y(1-y)(-c_y) \lt 0\) such that in both species cooperators dwindle and eventually go extinct. Interestingly, this happens regardless of the magnitude of foregone benefits, \(b_x,b_y\). Thus, unsurprisingly the demise of cooperation in inter-species interactions remains unchanged. | ||
For simplicity we assume that species \(X\) and \(Y\) face the same donation game, with costs \(c=c_x=c_y\) and benefits \(b=b_x=b_y\). Thus, \(X\) and \(Y\) are interchangeable labels | For simplicity we assume that species \(X\) and \(Y\) face the same donation game, with costs \(c=c_x=c_y\) and benefits \(b=b_x=b_y\). Thus, \(X\) and \(Y\) are interchangeable labels. | ||
== Spatial populations == | == Spatial populations == | ||
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# for \(r \to r_1\) from below, cooperation goes extinct in both layers simultaneously following the directed percolation universality class. Surprisingly, the extinction threshold \(r_1\approx0.02283(3)\) is very similar to the threshold in single species interactions \(r\approx0.0261\) (estimated from [[#References|Szabó & Hauert, 2002]]). Even though conditions for cooperation would seem much more challenging in interactions across species. | # for \(r \to r_1\) from below, cooperation goes extinct in both layers simultaneously following the directed percolation universality class. Surprisingly, the extinction threshold \(r_1\approx0.02283(3)\) is very similar to the threshold in single species interactions \(r\approx0.0261\) (estimated from [[#References|Szabó & Hauert, 2002]]). Even though conditions for cooperation would seem much more challenging in interactions across species. | ||
# spontaneous symmetry-breaking arises in the frequency of cooperation on the two layers when \(r \to r_2\). In fact, two equivalent asymmetric phases exist where individuals in one layer prey on those in the other layer or vice versa. This phase transition is similar to the spontaneous symmetry breaking in the two-dimensional Ising model in the absence of an external magnetic field. Although, the additional degrees of freedom (four instead of two microscopic states) result in deviations in the fluctuations of cooperation frequencies. | # spontaneous symmetry-breaking arises in the frequency of cooperation on the two layers when \(r \to r_2\). In fact, two equivalent asymmetric phases exist where individuals in one layer prey on those in the other layer or vice versa. This phase transition is similar to the spontaneous symmetry breaking in the two-dimensional Ising model in the absence of an external magnetic field. Although, the additional degrees of freedom (four instead of two microscopic states) result in deviations in the fluctuations of cooperation frequencies. | ||
# The third critical transition occurs for \(r \to r_3\) from above and is driven by bursts of \(DD\) domains. For small \(r\) the magnitude of the symmetry breaking decreases with \(r\), while the magnitude of bursts increases and causes fluctuations to diverge. However, the identification of the universal features of bursts and the slow transition towards absorbing states requires further, time-consuming simulations and analytical investigations. | # The third critical transition occurs for \(r \to r_3\) from above and is driven by bursts of \(DD\) domains. For small \(r\) the magnitude of the symmetry breaking decreases with \(r\), while the magnitude of bursts increases and causes fluctuations to diverge. However, the identification of the universal features of bursts and the slow transition towards absorbing states requires further, time-consuming simulations and analytical investigations. | ||
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==Publications== | ==Publications== | ||
# Hauert, C. & Szabó, G. (2024) Spontaneous symmetry breaking of cooperation between species, ''PNAS Nexus'' ( | # Hauert, C. & Szabó, G. (2024) Spontaneous symmetry breaking of cooperation between species, ''PNAS Nexus'' '''3''' (9) pgae326 doi: [https://doi.org/10.1093/pnasnexus/pgae326 10.1093/pnasnexus/pgae326] | ||
===References=== | ===References=== | ||