A study on H. magnifica and A. chrysopterus

After reading about various models aimed at analyzing mutualism in general, I decided to take some time and read studies that have been performed on anemones and anemonefish so that I can further understand their relationship and more appropriately choose a modeling approach. This study, conducted by  S.J. Holbrook and R.J. Schmitt in Moorea, French Polynesia over a three-year period and initiated in August 1999, provided much evidence supporting the ways in which anemones and anemonefish benefit one another, specifically how anemonefish affect anemones' growth, reproduction, and survival. 

• Anemones that harbored anemonefish grew nearly 3x faster than those that did not.
• The average daily growth increment was similar for those anemones that hosted 1 or 2 adult anemonefish, and both these growth rates were significantly greater than for anemones that lacked fish.
• Asexual reproduction was enhanced by the presence of anemonefish: anemones occupied by 2 fish underwent fission much more often (~2x expected rate) than those containing 1 (~expected rate) or 0 fish (~ ½  expected rate)
• Anemones that harbored fish experienced a much lower-than-expected probability of dying than those lacking fish: of 14 deaths, 11 were anemones without fish, 2 were anemones with 1 fish, and just 1 was an anemone with 2 fish.

Based on the models I have come across so far in my research, this information will help greatly. I have encountered many model equations containing parameters such as a, the factor by which one mutualist benefits the other. Now, at least for the benefits received by the anemone, I know that the presence of an anemonefish not only raises d, the mortality rate, but it also increases r, some growth rate, and b, some 'birth' rate. 

The authors point out that anemones could potentially derive both direct and indirect nutritive benefits from fish: anemones might directly ingest particles dropped by the fish or absorb their wastes, which could provide sources of regenerated nitrogen, sulfur, and phosphorous in addition to small prey and plankton they capture. And for at least some species of anemone, adult anemonefish defend their host anemone from specialized tentacle-eating fish predators such as Butterflyfish.

The authors also presented ways in which anemones appear to aid their resident anemonefish:

• Sea anemones provide an enemy-free space for anemonefish from its predators.
• The anemone acts as a nest site for an anemonefish: the anemonefish lay their eggs on the hard substrate beneath the oral disc of the anemone, where they are tended by the male fish.

As of now, I believe that anemonefish are obligate mutualists of anemones and thus cannot live without a host anemone, while anemones are merely facultative mutualists with anemonefish. Since there exists facultative, obligate, and obligate-with-thresholds mutualism, it will be interesting to determine where on this spectrum the anemone-anemonefish mutualism lies.

Differential equations 101

As I continue reading through papers written on mutualism and modeling in general, I am kicking myself for not having yet taken Ordinary Differential Equations. So I'm pausing for a bit to teach myself, at least the basics, of differential equations.

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.

Fitness of symbiont strategists

Since SpringerLink is having technical difficulties, I have not yet been able to read a paper that directly corrects Dean's, so I kept myself busy with this paper instead: Evolution of Marine Symbiosis--A Simple Cost-Benefit Model written by Jonathan Roughgarden and published in Ecology in 1975. 

Roughgarden took a completely different approach to modeling an example of symbiosis; he studied the fitness of the organisms by using the following parameters:

• Wss: fitness of an individual who has not yet attempted to colonize a host and is surviving as a free-living individual
• P: probability that an individual of symbiont phenotype successfully finds a host
• L: probability that the host survives while the symbiont is associated with it
• Wa: fitness of the symbiotic individual who has successfully found a host and is associated with it
• Wsg: fitness of the symbiotic individual who has failed to find a host or whose host has died

He then assumed that the search for a host involves some cost such as "passing up suitable sites for a solitary dwelling, devoting energy for the search which would otherwise be used in nest construction, and increased exposure to predation hazard." He determined relationships between the fitness of an individual and the cost of finding a host and decided that the expected fitness of a symbiont strategist (or potentially clownfish) was PLWa + (1-PL)Wsg. 

Roughgarden pointed out that for symbiosis to evolve, three factors must occur:

1. the host should be easy to find
2. the host should survive well with the symbiont
3. the host should provide substantial benefit to the symbiont

At this point, it seems as though there are many very different approaches I could take to modeling the relationship between clownfish and anemones, and it will be interesting to see how I end up utilizing and possibly combining some of the models I have encountered.

Mutualism models

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.

Preliminary research

Mutualism is defined as the interaction of two species of organisms that benefits both. Obligate mutualists survive only by association; facultative mutualists, while benefitting from the presence of one another, may also survive in the absence of each other. I am beginning preliminary research to further understand the mutualistic relationship between clownfish and sea anemones. 

Several theories exist as to the ways in which these organisms help one another:

• The territorial clownfish protect the anemone from anemone-eating fish.
• The stinging tentacles of the anemone protect the clownfish from its predators, while a mucus layer on the clownfish protects it from the stinging tentacles.
• The clownfish lures in other fish that are stung and consumed by the anemone.
• The clownfish increases oxygenation around the anemone and removes its waste material.

I am interested in determining through mathematical modeling whether clownfish and sea anemones are facultative, obligate, or obligate-with-thresholds mutualists and whether or not their mutualism is symmetric. I will also investigate how the size of the anemone and the number of clownfish residing within the anemone's tentacles affect each other.

To be determined:

• which species of clownfish to focus on: Amphiprion ocellaris (common clownfish) or Amphiprion percula (true clownfish)
• which species of anemone to focus on: Heteractis magnifica (Ritteri anemone) or Stichodactyla gigantea (giant carpet anemone) 
• what type of model to implement