Editing Theoretical ecology (section)

===Predator–prey interaction===
[[Predation|Predator–prey]] interactions exhibit natural oscillations in the populations of both predator and the prey.<ref name="BonsallHassell" /> In 1925, the American mathematician [[Alfred J. Lotka]] developed simple equations for predator–prey interactions in his book on biomathematics.<ref>Lotka,  A.J., ''Elements of Physical Biology'', [[Williams and Wilkins]], (1925)</ref> The following year, the Italian mathematician [[Vito Volterra]], made a statistical analysis of fish catches in the Adriatic<ref name="Goelmany">Goel, N.S. et al., "On the Volterra and Other Non-Linear Models of Interacting Populations", ''Academic Press Inc.'',  (1971)</ref> and independently developed the same equations.<ref>{{cite journal | last1 = Volterra | first1 = V | year = 1926 | title = Variazioni e fluttuazioni del numero d'individui in specie animali conviventi | journal = [[Accademia dei Lincei|Mem. Acad. Lincei Roma]] | volume = 2 | pages = 31–113 }}</ref> It is one of the earliest and most recognised ecological models, known as the [[Lotka–Volterra equation|Lotka-Volterra model]]:

:<math> \frac{dN(t)}{dt} = N(t)(r-\alpha P(t)) </math>
:<math> \frac{dP(t)}{dt} = P(t)(c \alpha N(t) -d) </math>
where N is the prey and P is the predator population sizes, r is the rate for prey growth, taken to be exponential in the absence of any predators, α is the prey mortality rate for per-capita predation (also called ‘attack rate’), c is the efficiency of conversion from prey to predator, and d is the exponential death rate for predators in the absence of any prey.

Volterra originally used the model to explain fluctuations in fish and shark populations after [[fishery|fishing]] was curtailed during the [[First World War]].  However, the equations have subsequently been applied more generally.<ref>{{cite book | last1=Begon | first1=M. | last2=Harper | first2=J. L. | last3=Townsend | first3=C. R. | year=1988 | title=Ecology: Individuals, Populations and Communities | publisher=Blackwell Scientific Publications Inc., Oxford, UK | title-link=Ecology: Individuals, Populations and Communities }}</ref> Other examples of these models include the Lotka-Volterra model of the [[snowshoe hare]] and [[Canadian lynx]] in North America,<ref>{{cite journal |author = C.S. Elton |year = 1924 |title = Periodic fluctuations in the numbers of animals - Their causes and effects  |journal = Journal of Experimental Biology  |volume = 2 |issue = 1 |pages = 119–163 |doi = 10.1242/jeb.2.1.119 |url=http://jeb.biologists.org/content/2/1/119.short|doi-access = free }}</ref> any infectious disease modeling such as the recent outbreak of [[SARS]]
<ref>{{cite journal |vauthors=Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S, James L, Gopalakrishna G, Chew SK, Tan CC, Samore MH, Fisman D, Murray M |year = 2003 |title = Transmission dynamics and control of severe acute respiratory syndrome |journal = Science |volume = 300 |issue = 5627 |pages = 1966–70 |doi = 10.1126/science.1086616 |pmid = 12766207 |pmc = 2760158|bibcode = 2003Sci...300.1966L }}</ref>
and biological control of [[California red scale]] by the introduction of its [[parasitoid]], ''[[Aphytis melinus]]''.<ref>{{cite journal |author1=John D. Reeve |author2=Wiliam W. Murdoch |year = 1986 |title = Biological Control by the Parasitoid Aphytis melinus, and Population Stability of the California Red Scale  |journal = Journal of Animal Ecology  |volume = 55 |issue = 3 |pages = 1069–1082 |doi = 10.2307/4434 |jstor = 4434
|bibcode=1986JAnEc..55.1069R }}</ref>

A credible, simple alternative to the Lotka-Volterra predator–prey model and their common prey dependent generalizations is the ratio dependent or [[Arditi-Ginzburg equations|Arditi-Ginzburg model]].<ref>{{cite journal | last1 = Arditi | first1 = R. | last2 = Ginzburg | first2 = L.R. | year = 1989 | title = Coupling in predator–prey dynamics: ratio dependence | url = http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf | journal = Journal of Theoretical Biology | volume = 139 | issue = 3 | pages = 311–326 | doi = 10.1016/s0022-5193(89)80211-5 | bibcode = 1989JThBi.139..311A | access-date = 2013-06-26 | archive-date = 2016-03-04 | archive-url = https://web.archive.org/web/20160304053545/http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf | url-status = dead }}</ref> The two are the extremes of the spectrum of predator interference models.  According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong.  They are much closer to the ratio-dependent extreme, so if a simple model is needed one can use the Arditi–Ginzburg model as the first approximation.<ref>Arditi, R. and Ginzburg, L.R. (2012) [https://books.google.com/books?id=c2m6XspGs-cC&dq=%22How+Species+Interact:+Altering+the+Standard+View+on+Trophic+Ecology%22&pg=PP2 ''How Species Interact: Altering the Standard View on Trophic Ecology''] Oxford University Press. {{ISBN|9780199913831}}.</ref>