Mutualism (biology) - Mathematical Modeling

Mathematical Modeling

In 1989, David Hamilton Wright developed a mathematical explanation for mutualism using the Lotka–Volterra equation. Wright modified the Lotka-Volterra equations by adding a new term, βM/K, to represent a mutualistic relationship.

The mutalistic relationship is quantified by:

where,

• N and M = the population density
• r = intrinsic growth rate of the population
• K = carrying capacity of its local environmental setting.
• β = coefficient converting encounters with one species to new units of the other

Mutualism is in essence the logistic growth equation + mutualistic interaction. The mutualistic interaction term represents the increase in population growth of species one as a result of the presence of greater numbers of species two, and vice versa. Wright also considered the concept of saturation, which means that with higher densities, there are decreasing benefits of further increases of the mutualist population. Without saturation, species' densities would increase indefinitely. Because that isn't 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