Mathematical treatments of mutualisms, like the study of mutualisms in general, have lagged behind those for
predation, or predator-prey, consumer-resource, interactions. In models of mutualisms, the terms "type I" and "type II"
functional responses refer to the linear and saturating relationships, respectively, between the
benefit provided to an individual of species 1 (
dependent variable) and the
density of species 2 (independent variable).
Type I functional response One of the simplest frameworks for modeling species interactions is the
Lotka–Volterra equations. In this model, the changes in population densities of the two mutualists are quantified as: : \begin{align} \frac{dN_1}{dt} &=r_1 N_1 - \alpha_{11} N_1^2 + \beta _{12}N_1N_2 \\[8pt] \frac{dN_2}{dt} &=r_2 N_2 - \alpha_{22} N_2^2 + \beta _{21}N_1N_2 \end{align} where • N_i = the population density of species
i. • r_i = the intrinsic growth rate of the population of species
i. • \alpha _{ii} = the negative effect of within-species crowding on species
i. • \beta _{ij} = the beneficial effect of the density of species
j on species
i. Mutualism is in essence the
logistic growth equation modified for mutualistic interaction. The mutualistic interaction term represents the increase in population growth of one species as a result of the presence of greater numbers of another species. As the mutualistic interactive term β is always positive, this simple model may lead to unrealistic unbounded growth. So it may be more realistic to include a further term in the formula, representing a saturation mechanism, to avoid this occurring.
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. 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. ;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
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 ::\cfrac{ax}{1+axT_H} where •
a = the instantaneous discovery rate •
x = food item density •
TH = handling time The equation that incorporates Type II functional response and mutualism is: : \frac{dN}{dt}=N\left[r(1-cN)+\cfrac{baM}{1+aT_H M}\right] 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 equation. •
a = instantaneous discovery rate •
b = coefficient converting encounters with
M to new units of
N or, equivalently, : \frac{dN}{dt}=N[r(1-cN)+\beta M/(X+M)] where •
X = 1/
aTH •
β =
b/
TH 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
isoclines 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 \cfrac{baM}{1+aT_H M}
vs. M. == Structure of networks ==