Editing Theoretical ecology (section)

==Modelling approaches==

As in most other sciences, mathematical models form the foundation of modern ecological theory.
* Phenomenological models: distill the functional and distributional shapes from observed patterns in the data, or researchers decide on functions and distribution that are flexible enough to match the patterns they or others (field or experimental ecologists) have found in the field or through experimentation.<ref name=Bolker>Bolker BM (2008) [https://books.google.com/books?id=yULf2kZSfeMC&pg=PA6 ''Ecological models and data in R''] Princeton University Press, pages 6–9. {{ISBN|978-0-691-12522-0}}.</ref>
* Mechanistic models: model the underlying processes directly, with functions and distributions that are based on theoretical reasoning about ecological processes of interest.<ref name=Bolker />

Ecological models can be [[Deterministic model|deterministic]] or [[stochastic]].<ref name=Bolker />

* Deterministic models always evolve in the same way from a given starting point.<ref>{{cite journal | vauthors = Sugihara G, May R | year = 1990 | title = Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series | url = http://deepeco.ucsd.edu/~george/publications/90_nonlinear_forecasting.pdf | journal = Nature | volume = 344 | issue = 6268 | pages = 734–41 | doi = 10.1038/344734a0 | pmid = 2330029 | bibcode = 1990Natur.344..734S | s2cid = 4370167 | access-date = 2011-05-13 | archive-url = https://web.archive.org/web/20110814112700/http://deepeco.ucsd.edu/~george/publications/90_nonlinear_forecasting.pdf | archive-date = 2011-08-14 | url-status = dead }}</ref> They represent the average, expected behavior of a system, but lack [[random variation]]. Many [[system dynamics]] models are deterministic.
* Stochastic models allow for the direct modeling of the random perturbations that underlie real world ecological systems. [[Markov process|Markov chain models]] are stochastic.

Species can be modelled in continuous or [[discrete time]].<ref name=Soetaert>Soetaert K and Herman PMJ (2009) [https://books.google.com/books?id=aVjDtSmJqhAC&dq=%22continuous+time%22+%22discrete+time%22+ecological&pg=PA273 ''A practical guide to ecological modelling''] Springer. {{ISBN|978-1-4020-8623-6}}.</ref>

* Continuous time is modelled using [[differential equation]]s.
* Discrete time is modelled using [[difference equation]]s. These model ecological processes that can be described as occurring over discrete time steps. [[Matrix (mathematics)|Matrix algebra]] is often used to investigate the evolution of age-structured or stage-structured populations. The [[Leslie matrix]], for example, mathematically represents the discrete time change of an age structured population.<ref>Grant WE (1986) ''Systems analysis and simulation in wildlife and fisheries sciences.'' Wiley, University of Minnesota, page 223. {{ISBN|978-0-471-89236-6}}.</ref><ref>Jopp F (2011) [https://books.google.com/books?id=AEgIo-BOzF0C&dq=%22Matrix+algebra%22+%22Leslie+matrix%22+ecological&pg=PA122 ''Modeling Complex Ecological Dynamics''] Springer, page 122. {{ISBN|978-3-642-05028-2}}.</ref><ref>Burk AR (2005) [https://books.google.com/books?id=B6qsUxgxcb8C&dq=%22Matrix+algebra%22+%22Leslie+matrix%22+ecological&pg=PA136 ''New trends in ecology research''] Nova Publishers, page 136. {{ISBN|978-1-59454-379-1}}.</ref>

Models are often used to describe real ecological reproduction processes of single or multiple species.
These can be modelled using stochastic [[branching process]]es. Examples are the dynamics of interacting populations ([[Theoretical ecology#Predator–prey interaction|predation]] [[Theoretical ecology#Competition .26 mutualism|competition and mutualism]]), which, depending on the species of interest, may best be modeled over either continuous or discrete time. Other examples of such models may be found in the field of [[Epidemic model|mathematical epidemiology]] where the dynamic relationships that are to be modeled are [[Theoretical ecology#Host–pathogen interaction|host–pathogen]] interactions.<ref name=Soetaert />

[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|left|Bifurcation diagram of the logistic map]]

[[Bifurcation theory]] is used to illustrate how small changes in parameter values can give rise to dramatically different long run outcomes, a mathematical fact that may be used to explain drastic ecological differences that come about in qualitatively very similar systems.<ref>Ma T and Wang S (2005) [https://books.google.com/books?id=dM6fVY65hHsC&q=%22Bifurcation+theory%22 ''Bifurcation theory and applications''] World Scientific. {{ISBN|978-981-256-287-6}}.</ref> [[Logistic map]]s are [[polynomial map]]pings, and are often cited as providing archetypal examples of how [[chaos theory|chaotic behaviour]] can arise from very simple [[non-linear]] dynamical equations. The maps were popularized in a seminal 1976 paper by the theoretical ecologist [[Robert May, Baron May of Oxford|Robert May]].<ref>{{cite book|first=Robert|last=May|author-link=Robert May, Baron May of Oxford|year=1976|title=Theoretical Ecology: Principles and Applications|publisher=Blackwell Scientific Publishers|isbn=978-0-632-00768-4}}</ref> The difference equation is intended to capture the two effects of reproduction and starvation.

In 1930, [[R.A. Fisher]] published his classic ''[[The Genetical Theory of Natural Selection]]'', which introduced the idea that frequency-dependent fitness brings a strategic aspect to [[evolution]], where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable offspring.<ref>{{cite book  |title=The genetical theory of natural selection  |last=Fisher  |first= R. A.  |year=1930 |location= Oxford   |publisher= The Clarendon press  |url=https://archive.org/details/geneticaltheoryo031631mbp }}</ref> In 1961, [[Richard Lewontin]] applied game theory to evolutionary biology in his ''Evolution and the Theory of Games'',<ref>{{cite journal |author = R C Lewontin |year = 1961 |title = Evolution and the theory of games |journal = Journal of Theoretical Biology |volume = 1 |issue = 3 |pages = 382–403 |doi = 10.1016/0022-5193(61)90038-8|pmid = 13761767 |bibcode = 1961JThBi...1..382L }}</ref>
followed closely by [[John Maynard Smith]], who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting",<ref>{{cite journal |author = John Maynard Smith |year = 1974 |title = Theory of games and evolution of animal conflicts |journal = Journal of Theoretical Biology |volume =47 |issue =1 |pages = 209–21 |doi = 10.1016/0022-5193(74)90110-6 |pmid = 4459582|bibcode = 1974JThBi..47..209M |url = http://www.dklevine.com/archive/refs4448.pdf }}</ref> defined the concept of the [[evolutionarily stable strategy]].

Because ecological systems are typically [[Nonlinearity|nonlinear]], they often cannot be solved analytically and in order to obtain sensible results, nonlinear, stochastic and computational techniques must be used. One class of computational models that is becoming increasingly popular are the [[agent-based model]]s. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in the real world. For example, optimal harvesting theory draws on optimization techniques developed in economics, computer science and operations research, and is widely used in [[fisheries]].<ref>Supriatna AK (1998) [https://books.google.com/books?id=njEoLwAACAAJ&q=%22optimal+harvesting+theory%22 ''Optimal harvesting theory for predator–prey metapopulations''] University of Adelaide, Department of Applied Mathematics.</ref>