Doebeli & Knowlton (1998) The evolution of interspecific mutualisms

Ref: Doebeli, M., & Knowlton, N. (1998). The evolution of interspecific mutualisms. Proc Natl Acad Sci USA, 95, 8676-8680. 

They tackle the evolution of mutualism between species. They posit that previous models which used a prisoner's dilemma made two assumptions that are inconsistent with the biology of most interspecific mutualism:
1. That players compete directly with their partners, and
2. That payoffs associated with cooperation, defection, etc. are constants.

Their model: 

Investments are variable. If a player invests I, it incurs a cost C(I) to itself and provides a benefit B(I) to its partner. Failure to invest means no cost or benefit.

Each host and symbiont is characterized by two parameters, (a, b), where a is the initial investment and b is the slope at which investments increase as an individual receives higher payoffs (see Fig 1.)

 Evolution (non-spatial): Hosts and symbionts are assigned a starting phenotype (a, b) and tested against mutants each generation. If the mutant host (symbiont) does better that the existing host (symbiont) against the existing symbiont (host), it is replaced.

Results (non-spatial): If b is held at zero, initial offers go monotonically to zero. If b can evolve, they also go to zero, but there are transient periods where mutualism increases, and average payoffs increase. This phenomenon led them to consider a spatial version.

The model (spatial extension): individuals situated on a dual lattice. Hosts and symbionts in same location interacted. Each type then competed with 8 nearest neighbors for reproduction.

Different versions:

• Stochastic competition (fig 3B): The occupant of each site was left unchanged with some probability.
• Stochastic payoffs (fig 3D): the actual benefit from investment I was drawn from a normal distribution with mean B(I).
• Asymmetric generation times (figs 3C and 3D): 100 symbiont generations per host generation.   

Results and interpretation: Both types of stochasticity were highly beneficial for the evolution of mutualism (Fig 3B and 3D). Our interpretation of this result is that stochasticity generates many more of the instances of transient local selection for more mutualistic phenotypes that lead to the overall maintenance of mutualism. Also unexpected was the effect of asymmetries in evolutionary rate, which were modeled by giving hosts and symbionts different values for generation time, mutation rate, or mutation magnitude (Fig. 3C). Without exception, the partner with the higher evolutionary rate [probably the smaller symbiont in many natural systems] had a higher investment and a lower payoff, a result consistent with otherwise puzzling patterns of unequal relative benefits in invertebrate – algal and lichen mutualisms.

Other results: