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. | |||
\begin{equation} | |||
\label{eq:dgr} | |||
\tag{2} | |||
{\bf A}= | |||
\begin{pmatrix} 1-r & -r\ | |||
1 & 0 \end{pmatrix}, | |||
\end{equation} | |||
where | |||
\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. | Eq. | ||
For simplicity we assume that species | For simplicity we assume that species | ||
== Spatial populations == | == Spatial populations == |
Revision as of 02:29, 21 February 2025
In mutualistic associations two species cooperate by exchanging goods or services with members of another species for their mutual benefit. At the same time competition for reproduction primarily continues with members of their own species. In intra-species interactions the prisoner's dilemma is the leading mathematical metaphor to study the evolution of cooperation. Here we consider inter-species interactions in the spatial prisoner's dilemma, where members of each species reside on one lattice layer. Cooperators provide benefits to neighbouring members of the other species at a cost to themselves. Hence, interactions occur across layers but competition remains within layers. We show that rich and complex dynamics unfold when varying the cost-to-benefit ratio of cooperation,
The problem of cooperation
The evolution and maintenance of cooperation ranks among the most fundamental questions in biological, social and economical systems. Cooperators provide benefits to others at a cost to themselves. Cooperation is a conundrum because on the one hand everyone benefits from mutual cooperation but on the other hand individuals face the temptation to increase their personal gains by defecting and withholding cooperation. This generates a conflict of interest between the group and the individual, which is termed a social dilemma. The prisoner's dilemma is the most popular mathematical metaphor of a social dilemma and theoretical tool to study cooperation through evolutionary game theory.
Donation game
In the simplest and most intuitive instance of the prisoner's dilemma, the donation game, two players meet and decide whether to cooperate and provide a benefit
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.
where
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.
Here we focus on the simplest setup with two identical species of equal population size each following the same ecological and evolutionary updating process.
Well-mixed populations
In intra-species interactions with random interactions, a so-called well-mixed population, the payoffs to cooperators and defectors soleley depends on their frequencies. In this case the replicator equation describes the change in the frequency of cooperators,
where
In inter-species interactions it is of crucial importance to distinguish between cooperation -- costly acts of individuals that benefit others -- and mutualistic interactions, where both parties benefit from the interaction. For the inter-species donation game cooperators of one species,
where
Eq.
For simplicity we assume that species
Spatial populations

In order to delineate effects of spatial dimensions on cooperation in mutualistic interactions, we consider two parallel, square lattice layers with
Dynamical domains
In spatial settings the formation of clusters reduces exploitation and increases interactions with other cooperators to make up for losses against defectors. Spatial assortment enables co-existence. This results in four dynamical domains depending on the cost-to-benefit ratio of cooperation,
: defection dominates. For high costs or low benefits, spatial correlations are insufficient to support cooperation and both populations evolve towards the absorbing state. The outcome is the same as in well-mixed populations, see . : cooperators and defectors co-exist. The frequency of cooperators in the dynamical equilibrium is equal in both layers and increases for decreasing . : spontaneous symmetry breaking. Cooperators and defectors continue to co-exist in a dynamical equilibrium but with essentially complementary strategy frequencies in the two layers. : relaxation into asymmetric absorbing states or . The emergence, growth and elimination of homogeneous domains of defection ( regions) drive spikes (or bursts) in defector frequencies.
The four dynamical regimes are separated by three different types of critical phase transitions.
Symmetric cooperation: random walk
The spatial configuration of cooperators,
The spatial configuration of cooperators,
Asymmetric cooperation
Typical spatial strategy distribution in the asymmetric phase with essentially complementary frequencies and distributions of cooperators and defectors in the two layers. Parameters are the same as in the previous figures but with
Bursts of defection
Snapshots a-d illustrate the spatial distribution of the four strategy pairs between lattices,
First, small islands of
Conclusions & Discussion
Interactions between species play a crucial role in symbiotic associations. More specifically, in mutualistic exchanges members of one species provide benefits in the form of goods or services to members of another species and vice versa. For costly provisions this results in an inter-species donation game but with a more fragile arrangement than for the standard, intra-species donation game. Here we studied the evolution of costly acts of cooperation on a two-layer square lattice, where each layer represents one species. This models a basic two-species ecological system. Evolution is controlled by increased reproduction or imitation of fitter neighbours within the same layer, while the payoff (or fitness) of individuals is determined through interactions between layers.
Using numerical simulations, we determined the frequency and fluctuation of cooperation in both layers as a function of the cost-to-benefit ratio,
- for
from below, cooperation goes extinct in both layers simultaneously following the directed percolation universality class. Surprisingly, the extinction threshold is very similar to the threshold in single species interactions (estimated from 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
. 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
from above and is driven by bursts of domains. For small the magnitude of the symmetry breaking decreases with , 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.
From a dynamical systems perspective this last critical transition is the most intriguing feature of our model. It poses a challenge for understanding the underlying mechanisms that characterize this universality class and drive the observed power law behaviour. Conversely, from a more applied, biological or social perspective, the spontaneous symmetry breaking and the dynamics of the asymmetric phase are equally clearly the most relevant and compelling feature. In particular, the mechanism for generating and maintaining the asymmetric phase appears to be intrinsically linked to the spikes in defection observed for
Symmetry breaking in the frequencies of cooperation between species has been reported before (see Ezoe & Ikegawa, 2013). However, the characteristics of the species association is not merely determined by the strategy frequencies but rather by the pairings of individual strategies across layers: mutualistic (
In the asymmetric phase, one species produces a social good at much higher rates than the other, which effectively separates the two species into producers and consumers. Interestingly, however, the average level of cooperation across both layers remains essentially constant at
Critical phase transitions

The social dilemma in Eq.
Each critical phase transition is characterized by, indeed defined by, diverging properties, such as fluctuations or correlation times and lengths. As a consequence it is numerically challenging to determine the exact transition points because the diverging quantities require both large lattices as well as long averaging times.
Overall, the frequency of cooperation tends to increase as the conditions of cooperation become more benign, i.e. lower costs or higher benefits, which translates into smaller ratios

In order to determine the boundaries
Note that
Directed percolation

Near the extinction threshold,
Coordination between lattices is almost perfect with very few
In agreement with theoretical expectations the critical transition at
When lowering the cost-to-benefit ratio,
Spontaneous symmetry breaking

Intriguingly, lowering
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
In the vicinity of the symmetry breaking transition,
Our analysis of the spontaneous symmetry breaking borrows concepts from the Ising model and critical phase transitions. The two-dimensional Ising model exhibits universal features through the power law scaling of the spontaneous magnetization in the absence of an external magnetic field, as well as the accompanying fluctuations when the temperature approaches the critical point. Here we use
However, the equivalence to the universal features of Ising type transitions is not obvious. Additional degrees of freedom result from the larger number of possible strategy pairs (
Bursts of defection

Finally, for
In particular, no theoretical arguments exist to date that explain and justify the observed power law behaviour of
This type of bursts are driven by sufficiently large spatial domains of an unstable phase. Evidently, their extension and frequency depends on
Another peculiarity of the burst dynamics is the feedback between their creation and extinction and the spatial configuration of the asymmetric phase: for
For
Publications
- Hauert, C. & Szabó, G. (2024) Spontaneous symmetry breaking of cooperation between species, PNAS Nexus (in print) doi: 10.1093/pnasnexus/pgae326
References
- Ezoe H. & Ikegawa Y. (2013) Coexistence of mutualists and non-mutualists in a dual-lattice model J. their. Biol. 332 1-8.
- Szabó, G. & Hauert, C. (2002) Phase Transitions and Volunteering in Spatial Public Goods Games, Phys. Rev. Lett. 89 (11) 118101.
- Hauert, C. & Doebeli, M. (2023) Spatial social dilemmas promote diversity Proc. Natl. Acad. Sci. USA 118 (42) e2105252118 doi: 10.1073/pnas.2105252118.