Editing Theoretical ecology (section)

=== Competition and mutualism ===
In studies of the populations of two species, the Lotka-Volterra system of equations has been extensively used to describe dynamics of behavior between two species, N<sub>1</sub> and N<sub>2</sub>.  Examples include relations between ''[[Dictyostelium discoideum|D. discoiderum]]'' and ''[[Escherichia coli|E. coli]]'',<ref>{{cite journal |author1=H. M. Tsuchiya |author2=J. F. Drake |author3=J. L. Jost |author4=A. G. Fredrickson  |name-list-style=amp |year = 1972 |title = Predator–Prey Interactions of ''Dictyostelium discoideum'' and ''Escherichia coli'' in Continuous Culture1 |journal = Journal of Bacteriology |volume = 110 |issue = 3 |pages = 1147–53 |doi=10.1128/JB.110.3.1147-1153.1972 |url= |pmid = 4555407 |pmc = 247538}}</ref>
as well as theoretical analysis of the behavior of the system.<ref>{{cite journal | title = Cooperative systems theory and global stability of diffusion models | journal = Acta Applicandae Mathematicae | volume = 14 | issue = 1–2| pages = 49–57 | doi = 10.1007/BF00046673 | year = 1989 | last1 = Takeuchi | first1 = Y. | s2cid = 189902519 }}</ref>
:<math> \frac{dN_1}{dt} = \frac{r_1N_1}{K_1}\left( K_1 - N_1 + \alpha_{12}N_2 \right) </math>
:<math> \frac{dN_2}{dt} = \frac{r_2N_2}{K_2}\left( K_2 - N_2 + \alpha_{21}N_1 \right) </math>
The r coefficients give a “base” growth rate to each species, while K coefficients correspond to the carrying capacity.  What can really change the dynamics of a system, however are the α terms.  These describe the nature of the relationship between the two species.  When α<sub>12</sub> is negative, it means that N<sub>2</sub> has a negative effect on N<sub>1</sub>, by competing with it, preying on it, or any number of other possibilities.  When α<sub>12</sub> is positive, however, it means that N<sub>2</sub> has a positive effect on N<sub>1</sub>, through some kind of mutualistic interaction between the two.
When both α<sub>12</sub> and α<sub>21</sub> are negative, the relationship is described as [[Biological competition|competitive]].  In this case, each species detracts from the other, potentially over competition for scarce resources.
When both α<sub>12</sub> and α<sub>21</sub> are positive, the relationship becomes one of [[mutualism (biology)|mutualism]].  In this case, each species provides a benefit to the other, such that the presence of one aids the population growth of the other.

:''See [[Competitive Lotka–Volterra equations]] for further extensions of this model.''