Editing Mutualism (biology) (section)

===Type II functional response===
In 1989, David Hamilton Wright modified the above Lotka–Volterra equations by adding a new term, ''βM''/''K'', to represent a mutualistic relationship.<ref name="Wright">{{cite journal | last1=Wright | first1=David Hamilton | year=1989 | title=A Simple, Stable Model of Mutualism Incorporating Handling Time | journal=[[The American Naturalist]] | volume=134 | issue=4| pages=664–7 | doi=10.1086/285003| bibcode=1989ANat..134..664W | s2cid=83502337 }}</ref> Wright also considered the concept of saturation, which means that with higher densities, there is a decrease in the benefits of further increases of the mutualist population. Without saturation, depending on the size of parameter α, species densities would increase indefinitely. Because that is not possible due to environmental constraints and carrying capacity, a model that includes saturation would be more accurate. Wright's mathematical theory is based on the premise of a simple two-species mutualism model in which the benefits of mutualism become saturated due to limits posed by handling time. Wright defines handling time as the time needed to process a food item, from the initial interaction to the start of a search for new food items and assumes that processing of food and searching for food are mutually exclusive. Mutualists that display foraging behavior are exposed to the restrictions on handling time. Mutualism can be associated with symbiosis.{{citation needed|date=June 2024}}

;Handling time interactions

In 1959, [[C. S. Holling]] performed his classic disc experiment that assumed that
# the number of food items captured is proportional to the allotted [[Search time|searching time]]; and
# that there is a [[handling time]] variable that exists separately from the notion of search time. He then developed an equation for the Type II [[functional response]], which showed that the feeding rate is equivalent to

::<math>\cfrac{ax}{1+axT_H}</math>

where
* ''a'' = the instantaneous discovery rate
* ''x'' = food item density
* ''T''<sub>''H''</sub> = handling time

The equation that incorporates Type II functional response and mutualism is:

:<math>
\frac{dN}{dt}=N\left[r(1-cN)+\cfrac{baM}{1+aT_H M}\right]
</math>

where
* ''N'' and ''M'' = densities of the two mutualists
* ''r'' = intrinsic rate of increase of ''N''
* ''c'' = coefficient measuring negative intraspecific interaction. This is equivalent to inverse of the [[carrying capacity]], 1/''K'', of ''N'', in the [[Logistic function#Logistic differential equation|logistic equation]].
* ''a'' = instantaneous discovery rate
* ''b'' = coefficient converting encounters with ''M'' to new units of ''N''

or, equivalently,

:<math>
\frac{dN}{dt}=N[r(1-cN)+\beta M/(X+M)]
</math>

where
* ''X'' = 1/''aT''<sub>H</sub>
* ''β'' = ''b''/''T''<sub>H</sub>

This model is most effectively applied to free-living species that encounter a number of individuals of the mutualist part in the course of their existences. Wright notes that models of biological mutualism tend to be similar qualitatively, in that the featured [[isocline]]s generally have a positive decreasing slope, and by and large similar isocline diagrams. Mutualistic interactions are best visualized as positively sloped isoclines, which can be explained by the fact that the saturation of benefits accorded to mutualism or restrictions posed by outside factors contribute to a decreasing slope.

The type II functional response is visualized as the graph of <math>\cfrac{baM}{1+aT_H M}</math> ''vs.'' ''M''.