Mutualisms: cooperation between species

Author: Christoph Hauert

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, $r$. Four distinct dynamical domains emerge that are separated by critical phase transitions, each characterized by diverging fluctuations in the frequency of cooperation: (i) for large $r$ cooperation is too costly and defection dominates; (ii) for lower $r$ cooperators survive at equal frequencies in both species; (iii) lowering $r$ further results in an intriguing, spontaneous symmetry breaking of cooperation between species with increasing asymmetry for decreasing $r$; (iv) finally, for small $r$, bursts of mutual defection appear that increase in size with decreasing $r$ and eventually drive the populations into absorbing states. Typically one species is cooperating and the other defecting and hence establish perfect asymmetry. Intriguingly and despite the symmetrical model setup, natural selection can nevertheless favour the spontaneous emergence of asymmetric evolutionary outcomes where, on average, one species exploits the other in a dynamical equilibrium.

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 $b$ to their interaction partner at a cost $c$ ($b>c>0$) or to defect, which entails no costs and provides no benefits. If both cooperate each gets $b-c$ but if one defects then the defector gets $b$ and shirks the costs of cooperation, while the cooperator is stuck with the costs $-c$. Finally if both defect, no one gets anything. The interaction is conveniently summarized in a payoff matrix (for the row player):

\begin{align} \label{eq:pd} \begin{matrix}

& \begin{matrix}C&D\end{matrix} \\\\

$$\begin{array}{c}C\\ \\ D\end{array}$$ &

 \begin{pmatrix}b-c\\\\-c\\\\b\\\\0\end{pmatrix}\\\\

\end{matrix} \end{align}

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.

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, $x$, in a population as a function of the expected payoffs of cooperators and defectors, ${\xi }_C$ and ${\xi }_D$, respectively:

$$\begin{array}{}\text{(1)}& \dot{x}& =x({\xi }_C-\overline{\xi })=x(1-x)({\xi }_C-{\xi }_D)\end{array}$$

where ${\xi }_C=xb-c$, ${\xi }_D=xb$, and $\overline{\xi }=x{\xi }_C+(1-x){\xi }_D$ denotes the average payoff in the population. Since $\dot{x}<0$ for all $x$, the frequency of cooperators decreases and eventually goes extinct. This is the well-known result that cooperation is not evolutionarily stable in the prisoner's dilemma.

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, $X$, provide benefits to members of another species, $Y$, at a cost to themselves. In well-mixed populations, where members of one species interact with random individuals of the other species, the corresponding two-species replicator equation, for the frequencies of cooperators $x$ in one species and $y$ in the other, is given by

$$\text{Label 'eq:replicator' multiply defined}$$

where ${\xi }_C=y{b}_x-c_x$ and ${\zeta }_C=x{b}_y-c_y$ denote the expected payoffs for cooperators in species $X$ and $Y$ with costs $c_x,c_y$ and benefits $b_x,b_y$, respectively. Defectors equally reap the benefits but bear no costs, ${\xi }_D=y{b}_x$ and ${\zeta }_D=x{b}_y$, respectively. Note, the benefits to either species depend on the frequency of cooperators in the other species, who also bears the costs of cooperation. The average payoff to each species is $\overline{\xi }=x{\xi }_C+y{\xi }_D=y{b}_x-x{c}_x$ and $\bar\zeta = x \zeta_C + y \zeta_D = x b_y-y c_y$, respectively.

Eq. $\text{(1)}$ reduces to $\dot{x}=x(1-x)(-c_x)<0$ and $\dot{y}=y(1-y)(-c_y)<0$ 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. Rescaling reduces the interaction in Eq. $\text{(???)}$ to a single parameter:

\begin{equation} {\bf A}=\begin{pmatrix} 1-r & -r \cr

                       1 & 0 \end{pmatrix}\,,

\label{eq:Adg} \end{equation}

where $r=c/b$ denotes the cost-to-benefit ratio with $0<r<1$.

Spatial populations

File:Mutualism two-layer lattice.svg

On lattices with $k=4$ neighbours, an individual in species $X$ ($\bullet$) interacts (dashed lines) with its $k+1$ neighbours in the other species $Y$ ($\bullet$ and $\bullet$) but competes (solid blue lines) with its $k$ neighbours in its own species $X$ ($\bullet$). Analogously, a member of species $Y$ ($\bullet$) interacts with their neighbours in species $X$ ($\bullet$ and $\bullet$) and competes with neighbours of their own species ($\bullet$).

In order to delineate effects of spatial dimensions on cooperation in mutualistic interactions, we consider two parallel, square lattice layers with $N=L\times L$ sites each, where $L$ refers to the linear lattice dimension. Each site is occupied by one individual with $k=4$ neighbours under periodic boundary conditions. One layer represents species $X$ and the other species $Y$: individuals in each layer interact with their $k+1$ nearest neighbours on the other layer but compete with $k$ neighbours in their own layer, see figure.

References

1. Hauert, C. & Szabó, G. (2024) Spontaneous symmetry breaking of cooperation between species, PNAS Nexus (in print)