A new phenomenon arises in equations for unrestrained mutualism, one not possible for competition or predation. It can be shown that unrestrained mutualists are pulled into fixed ratios which are based on how their interaction terms differ from their self-limiting terms. As populations grow and move toward these fixed ratios, the two equations governing the individual species collapse, in effect, into an equation of a single-species. This single species grows orthologistically, a form of growth you saw earlier in Figure 4.3. The system behaves mathematically as if only one species were participating.
What does the mathematical collapse to a single equation mean biologically? It suggests that the biological world could respond analogously—that two actual species growing as unrestrained mutualists could intermingle into a single quasi-species—at least into the best approximation of a single species that biology could accomplish using genetically different entities. Lichen and the eukaryotic cell are examples, genetically separate but biologically merged. Indeed, lichens were thought to be individual species until the nineteenth century, and the eukaryotic cell was only accepted as the result of mutualistic combinations late in the twentieth.
It therefore appears that natural selection has not overlooked this possibility. In a community model with mutualistic species locked in approach to a singularity, the two mutualists may be replaced in the model by a new quasi-species, representing the two species jointly but ultimately having a non-singular form of population growth. The resulting quasi-species may grow without obvious inhibition toward a singularity, then switch to a different model, as you have seen for single-species models in Figure 4.4 and for our own species in Figure 6.3.