A step in the right direction

Roughgarden presents a very interesting take on modeling mutualism. He begins by explaining that the initial formation of association occurs when the fitness of a symbiont strategy exceeds that of a free-living or solitary strategy. Then given established symbiosis, the issue of possible mutualistic behavior by the guest may be raised. 

A simple concept follows: assuming that the evolution of such mutualism passes through a parasitic phase, it turns on whether a guest who sacrifices some of the benefit he extracts from the host in order to improve his host's survival is fitter on average than a guest who extracts the maximum benefit from the host regardless

 of the consequences to the host's survival.

The sacrifice made by the mutualist (S) to improve its host's survival is profitable in terms of his own fitness if the maximum benefit which could be provided by an association (Bmax) plus the cost of finding a host (C) multiplied by the quantity of the difference in probability of survival between a host associated with a mutualistic guest (Lm) and one associated with a nonmutualistic guest (Lp) divided by the Lm is greater than the sacrifice. Then, the factors that promote mutualism are:

1. a host provides considerable improvement over a purely solitary existence;
2. a high dependency by the guest for the host as reflected in the loss of ability to survive in a solitary state resulting from an unsuccessful host search; and
3. a mutualistic behavior that does in fact give a large improvement to the host's survival, i.e., a behavior that makes (Lm -Lp) large

Since one condition states that mutualism will evolve only if the host survives poorly enough so that improvement is actually feasible, while another condition states that an association cannot form unless the host's survival is high, mutualism should be observed only on hosts of intermediate survival ability, as shown below.

A joint equilibrium can then be found by writing mutualist fitness as a function of S and C and fidning the maximum, which leads to two simultaneous quadratics whose solutions is unnecessary to the qualitative conclusion that the evolution of mutualism should lead to an association that is obligatory for the guest. 

To illustrate the use of his cost-benefit model, Roughgarden considered a study conducted by Verwey in 1930 regarding the associations between five species of damselfish (Pomicentridae) and five species of large sea anemones from the genus Stoichactis! I was not expecting this direct tie with what I am studying, so with the help of Earl Gregg Swem Library's ILLiad Interlibrary Loan System, I am off to find Verwey's article in Treubia: A Journal on Zoology and Hydrobiology of the Indo-Australian Archipelago.