I was surprised to find many papers on mutualism models, all of which cited A Simple Model of Mutualism written by Antony M. Dean and published in The American Naturalist in 1983. The main idea of his model was that all that is required for mutualism to occur "is that the number of one mutualist maintained by a certain number of the other mutualist be greater than the number of the former needed to maintain that certain number of the latter. If true, then both populations will grow until the density effects limit the growth of the carrying capacities so that isoclines 3 and 4 will inevitably intersect at a point of stable equilibrium," as shown in the first graph (a) above. The second graph (b) shows an unstable equilibrium when the isoclines touch; if environmental perturbation causes one or both species to be reduced in number such that the point falls into the hatched region, then both populations decline to extinction. The third graph (c) shows when the isoclines don’t intersect, and no mutualism may occur in (*b*) or (*c*).

Dean continued by pointing out that mutualisms must be stabilized by factors external to these simple models such as competition or predation by a third species, the introduction of rate-limiting resources, competition for rate-limiting resources, and inhibitory resources. He determined that stability arises from the fact that the carrying capacities of mutualist populations are dependent upon each other's abundance, showing diminishing returns as they increase. The populations are therefore self-limiting because members of each population compete for limited resources.

Some papers written since Dean's was published have critiqued his model and corrected singularities, so the next step is to not only examine the papers referenced by Dean himself but to follow up on what has been accomplished since 1983 and how Dean's model has been improved.

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