Editing Theoretical ecology

{{Short description|Scientific discipline}}
[[File:FoodWeb.svg|thumb|300px|right|Mathematical models developed in theoretical ecology predict complex [[food web]]s can be less stable than simpler webs.<ref name=May />{{rp|75–77}}<ref name=Pimm />{{rp|64}}]]
[[File:Fig.2 Life on Earth-Flow of Energy and Entropy.gif|thumb|300px|right|Life on Earth-Flow of Energy and Entropy]]
'''Theoretical ecology''' is the scientific discipline devoted to the study of [[ecosystem|ecological systems]] using theoretical methods such as simple [[conceptual model]]s, [[mathematical model]]s, [[computer simulation|computational simulation]]s, and advanced [[data analysis]]. Effective models improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.

The field is broad and includes foundations in applied mathematics, computer science, biology, statistical physics, genetics, chemistry, evolution, and conservation biology. Theoretical ecology aims to explain a diverse range of phenomena in the life sciences, such as population growth and [[population dynamics|dynamics]], fisheries, [[biological competition|competition]], evolutionary theory, epidemiology, animal behavior and group dynamics, [[food webs]], ecosystems, spatial ecology, and the [[effects of climate change]].

Theoretical ecology has further benefited from the advent of fast computing power, allowing the analysis and visualization of large-scale computational simulations of ecological phenomena.  Importantly, these modern tools provide [[Quantitative ecology|quantitative predictions]] about the effects of human induced environmental change on a diverse variety of ecological phenomena, such as: species invasions, climate change, the effect of fishing and hunting on food network stability, and the global [[carbon cycle]].

==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>

==Population ecology==
{{main|Population ecology}}
'''Population ecology''' is a sub-field of [[ecology]] that deals with the dynamics of [[species]] [[population]]s and how these populations interact with the [[natural environment|environment]].<ref name="Odum1959">{{cite book | last =Odum | first =Eugene P. | author-link =Eugene Odum | title =Fundamentals of Ecology | url =https://archive.org/details/fundamentalsofec0000odum/page/546 | url-access =registration | edition =Second | publisher =W. B. Saunders Co. | year =1959 | location =Philadelphia and London | isbn =9780721669410 | oclc =554879 | page =[https://archive.org/details/fundamentalsofec0000odum/page/546 546 p] }}</ref> It is the study of how the [[population size]]s of species living together in groups change over time and space, and was one of the first aspects of ecology to be studied and modelled mathematically.

===Exponential growth===
{{Main|Exponential growth}}

The most basic way of modeling population dynamics is to assume that the rate of growth of a population depends only upon the population size at that time and the per capita growth rate of the organism. In other words, if the number of individuals in a population at a time t, is N(t), then the rate of population growth is given by:
:<math> \frac{dN(t)}{dt}=rN(t) </math>
where r is the per capita growth rate, or the intrinsic growth rate of the organism.  It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This [[Linear differential equation#First order equation|first order]] [[linear differential equation]] can be solved to yield the solution
:<math> N(t) = N(0) \ e^{rt} </math>,
a trajectory known as [[Malthusian growth]], after [[Thomas Malthus]], who first described its dynamics in 1798. A population experiencing Malthusian growth follows an exponential curve, where N(0) is the initial population size. The population grows when r > 0, and declines when r < 0. The model is most applicable in cases where a few organisms have begun a colony and are rapidly growing without any limitations or restrictions impeding their growth (e.g. bacteria inoculated in rich media).

===Logistic growth===
{{Main|Logistic growth}}

The exponential growth model makes a number of assumptions, many of which often do not hold. For example, many factors affect the intrinsic growth rate and is often not time-invariant. A simple modification of the exponential growth is to assume that the intrinsic growth rate varies with population size. This is reasonable: the larger the population size, the fewer resources available, which can result in a lower birth rate and higher death rate. Hence, we can replace the time-invariant r with r’(t) = (b –a*N(t)) – (d + c*N(t)), where a and c are constants that modulate birth and death rates in a population dependent manner (e.g. [[intraspecific competition]]). Both a and c will depend on other environmental factors which, we can for now, assume to be constant in this approximated model. The differential equation is now:<ref name=Moss>Moss R, Watson A and Ollason J (1982) [https://books.google.com/books?id=l9YOAAAAQAAJ&dq=%22Logistic+growth%22&pg=PA52 ''Animal population dynamics''] Springer, page 52–54. {{ISBN|978-0-412-22240-5}}.</ref>
:<math> \frac{dN(t)}{dt}=((b-aN(t))-(d-cN(t)))N(t) </math>
This can be rewritten as:<ref name=Moss />
:<math> \frac{dN(t)}{dt}=rN(t) \left(1-\frac{N}{K}\right) </math>
where r = b-d and K = (b-d)/(a+c).

The biological significance of K becomes apparent when stabilities of the equilibria of the system are considered. The constant K is the [[carrying capacity]] of the population.  The equilibria of the system are N = 0 and N = K. If the system is linearized, it can be seen that N = 0 is an unstable equilibrium while K is a stable equilibrium.<ref name=Moss />

===Structured population growth===
{{See also|Matrix population models}}

Another assumption of the exponential growth model is that all individuals within a population are identical and have the same probabilities of surviving and of reproducing.  This is not a valid assumption for species with complex life histories.  The exponential growth model can be modified to account for this, by tracking the number of individuals in different age classes (e.g. one-, two-, and three-year-olds) or different stage classes (juveniles, sub-adults, and adults) separately, and allowing individuals in each group to have their own survival and reproduction rates.
The general form of this model is
:<math>\mathbf{N}_{t+1} = \mathbf{L}\mathbf{N}_t</math>
where '''N<sub>t</sub>''' is a [[Euclidean vector|vector]] of the number of individuals in each class at time ''t'' and '''L''' is a [[Matrix (mathematics)|matrix]] that contains the survival probability and fecundity for each class.  The matrix '''L''' is referred to as the [[Leslie matrix]] for [[Age class structure|age-structured]] models, and as the Lefkovitch matrix for [[Ontogeny|stage-structured]] models.<ref>{{cite book |author=Hal Caswell |title= Matrix Population Models: Construction, Analysis, and Interpretation |year= 2001 |publisher=Sinauer}}</ref>

If parameter values in '''L''' are estimated from demographic data on a specific population, a structured model can then be used to predict whether this population is expected to grow or decline in the long-term, and what the expected [[population pyramid|age distribution]] within the population will be.  This has been done for a number of species including [[loggerhead sea turtle]]s and [[right whale]]s.<ref>{{cite journal |author=D.T.Crouse, L.B. Crowder, H.Caswell |year=1987 |title=A stage-based population model for loggerhead sea turtles and implications for conservation  |journal=Ecology |pages=1412–1423 |doi=10.2307/1939225 |volume=68 |issue=5|jstor=1939225 |bibcode=1987Ecol...68.1412C |s2cid=16608658 }}</ref><ref>{{cite journal |author1=M. Fujiwara |author2=H. Caswell |year = 2001 |title = Demography of the endangered North Atlantic right whale  |journal = Nature  |volume = 414 |issue = 6863 |pages = 537–541 |doi=10.1038/35107054 |pmid = 11734852 |bibcode = 2001Natur.414..537F|s2cid=4407832 }}</ref>

==Community ecology==
{{main|Community ecology}}

An ecological community is a group of trophically similar, [[sympatric]] species that actually or potentially compete in a local area for the same or similar resources.<ref name=Hubbell2001/> Interactions between these species form the first steps in analyzing more complex dynamics of ecosystems.  These interactions shape the distribution and dynamics of species. Of these interactions, predation is one of the most widespread population activities.<ref name="BonsallHassell">{{cite book |last1= Bonsall |first1= Michael B. |last2= Hassell |first2= Michael P. |editor1-first= Robert |editor1-last= May |editor2-first= Angela |editor2-last= McLean |title= Theoretical Ecology: Principles and Applications |edition=3rd |year= 2007 |publisher= Oxford University Press |pages=46–61 |chapter=Predator–prey interactions}}</ref>
Taken in its most general sense, predation comprises predator–prey, host–pathogen, and host–parasitoid interactions.

[[File:Cheetah Baboon LV.jpg|thumb|right|Lotka–Volterra model of cheetah–baboon interactions. Starting with 80 baboons (green) and 40 cheetahs, this graph shows how the model predicts the two species numbers will progress over time.]]

===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>

===Host–pathogen interaction===
{{See also|Compartmental models in epidemiology}}

The second interaction, that of host and [[pathogen]], differs from predator–prey interactions in that pathogens are much smaller, have much faster generation times, and require a host to reproduce.  Therefore, only the host population is tracked in host–pathogen models.  Compartmental models that categorize host population into groups such as susceptible, infected, and recovered (SIR) are commonly used.<ref>{{cite book |last1= Grenfell |first1= Bryan |last2= Keeling |first2= Matthew |editor1-first= Robert |editor1-last= May |editor2-first= Angela |editor2-last= McLean |title= Theoretical Ecology: Principles and Applications |edition=3rd |year= 2007 |publisher= Oxford University Press |pages=132–147 |chapter=Dynamics of infectious disease}}</ref>

===Host–parasitoid interaction===
The third interaction, that of host and [[parasitoid]], can be analyzed by the [[Nicholson–Bailey model]], which differs from Lotka-Volterra and SIR models in that it is discrete in time.  This model, like that of Lotka-Volterra, tracks both populations explicitly.  Typically, in its general form, it states:
:<math> N_{t+1} = \lambda \ N_t \ [ 1 - f(N_t, P_t) ]</math>
:<math> P_{t+1} = c \ N_t \ f(N_t, p_t) </math>
where f(N<sub>t</sub>, P<sub>t</sub>) describes the probability of infection (typically, [[Poisson distribution]]), λ is the per-capita growth rate of hosts in the absence of parasitoids, and c is the conversion efficiency, as in the Lotka-Volterra model.<ref name="BonsallHassell" />

=== 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.''

===Neutral theory===
{{Main|Unified neutral theory of biodiversity}}

[[Unified neutral theory of biodiversity|Unified neutral theory]] is a hypothesis proposed by [[Stephen P. Hubbell]] in 2001.<ref name=Hubbell2001>{{Cite web
 |author       = Hubbell, SP
 |year         = 2001
 |title        = The Unified Neutral Theory of Biodiversity and Biogeography (MPB-32)
 |url          = https://pup.princeton.edu/chapters/s7105.html
 |access-date  = 2010-12-16
 |archive-url  = https://web.archive.org/web/20110718012434/https://pup.princeton.edu/chapters/s7105.html
 |archive-date = 2011-07-18
 |url-status     = dead
}}</ref> The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other [[Unified neutral theory of biodiversity|neutral theories]] in ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. Neutrality means that at a given [[trophic level]] in a [[food web]], species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis.<ref name=Hubbell2005>{{Cite journal | doi = 10.1666/0094-8373(2005)031[0122:TNTOBA]2.0.CO;2 | last1 = Hubbell | first1 = S. P. | title = The neutral theory of biodiversity and biogeography and Stephen Jay Gould | journal = Paleobiology | volume = 31 | pages = 122–123 | year = 2005| s2cid = 86271294 }}</ref>  This implies that biodiversity arises at random, as each species follows a [[random walk]].<ref name="McGill2003">{{Cite journal | last1 = McGill | first1 = B. J. | title = A test of the unified neutral theory of biodiversity | journal = Nature | volume = 422 | issue = 6934 | pages = 881–885 | year = 2003 | pmid = 12692564 | doi = 10.1038/nature01583| bibcode = 2003Natur.422..881M | s2cid = 1627734 }}</ref>  This can be considered a [[null hypothesis]] to [[ecological niche|niche theory]]. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.

Under unified neutral theory, complex ecological interactions are permitted among individuals of an [[ecological community]] (such as competition and cooperation), providing all individuals obey the same rules. Asymmetric phenomena such as [[parasite|parasitism]] and [[predator|predation]] are ruled out by the terms of reference; but cooperative strategies such as [[swarming]], and negative interaction such as competing for limited food or light are allowed, so long as all individuals behave the same way. The theory makes predictions that have implications for the management of [[biodiversity]], especially the management of rare species. It predicts the existence of a fundamental biodiversity constant, conventionally written ''θ'', that appears to govern species richness on a wide variety of spatial and temporal scales.

Hubbell built on earlier neutral concepts, including [[Robert H. MacArthur|MacArthur]] & [[E. O. Wilson|Wilson]]'s theory of [[island biogeography]]<ref name="Hubbell2001"/> and [[Stephen Jay Gould|Gould]]'s concepts of symmetry and null models.<ref name=Hubbell2005/>

==Spatial ecology==
{{Main|Spatial ecology}}

===Biogeography===
[[Biogeography]] is the study of the distribution of species in space and time. It aims to reveal where organisms live, at what abundance, and why they are (or are not) found in a certain geographical area.

Biogeography is most keenly observed on islands, which has led to the development of the subdiscipline of [[island biogeography]]. These habitats are often a more manageable areas of study because they are more condensed than larger ecosystems on the mainland. In 1967, [[Robert MacArthur]] and [[Edward Osborne Wilson|E.O. Wilson]] published ''[[The Theory of Island Biogeography]]''. This showed that the species richness in an area could be predicted in terms of factors such as habitat area, immigration rate and extinction rate.<ref>MacArthur RH and Wilson EO (1967) [https://books.google.com/books?id=a10cdkywhVgC&dq=biogeography&pg=PR7 ''The theory of island biogeography'']</ref> The theory is considered one of the fundamentals of ecological theory.<ref name="Wiens04">{{Cite journal |last1 = Wiens |first1 = J. J. |last2 = Donoghue |first2 = M. J. |title = Historical biogeography, ecology and species richness |journal = Trends in Ecology and Evolution |volume = 19 |issue = 12 |pages=639–644 |year = 2004 |url = http://www.phylodiversity.net/donoghue/publications/MJD_papers/2004/144_Wiens_TREE04.pdf |archive-url = https://web.archive.org/web/20100601214300/http://www.phylodiversity.net/donoghue/publications/MJD_papers/2004/144_Wiens_TREE04.pdf |url-status = usurped |archive-date = June 1, 2010 |doi = 10.1016/j.tree.2004.09.011 |pmid = 16701326 }}</ref> The application of island biogeography theory to [[habitat fragmentation|habitat fragments]] spurred the development of the fields of [[conservation biology]] and [[landscape ecology]].<ref>This applies to British and American academics; landscape ecology has a distinct genesis among European academics.</ref>

==== r/K-selection theory ====
{{Main|r/K selection}}

A population ecology concept is r/K selection theory, one of the first predictive models in ecology used to explain [[Life history theory|life-history evolution]]. The premise behind the r/K selection model is that natural selection pressures change according to [[population densities|population density]]. For example, when an island is first colonized, density of individuals is low. The initial increase in population size is not limited by competition, leaving an abundance of available [[Resource (biology)|resources]] for rapid population growth. These early phases of [[population growth]] experience ''density-independent'' forces of natural selection, which is called ''r''-selection. As the population becomes more crowded, it approaches the island's carrying capacity, thus forcing individuals to compete more heavily for fewer available resources. Under crowded conditions, the population experiences density-dependent forces of natural selection, called ''K''-selection.<ref>First introduced in MacArthur & Wilson's (1967) book of notable mention in the history and theoretical science of ecology, ''[[The Theory of Island Biogeography]]''</ref><ref name="Reznick2002">{{Cite journal |last1=Reznick |first1=D. |last2=Bryant |first2=M. J. |last3=Bashey |first3=F. |title=r- and K-selection revisited: The role of population regulation in life-history evolution |journal=Ecology |volume=83 |issue=6 |pages=1509–1520 |year=2002 |url=http://www2.hawaii.edu/~taylor/z652/Reznicketal.pdf |doi=10.1890/0012-9658(2002)083[1509:RAKSRT]2.0.CO;2 |issn=0012-9658 |access-date=2015-01-05 |archive-url=https://web.archive.org/web/20101230233401/http://www2.hawaii.edu/~taylor/z652/Reznicketal.pdf |archive-date=2010-12-30 |url-status=dead }}</ref>

[[File:Coral Reef aerial.png|thumb|300px|right|The diversity and containment of [[coral reef]] systems make them good sites for testing [[Environmental niche modelling|niche]] and [[Unified Neutral Theory of Biodiversity and Biogeography|neutral theories]].<ref>{{cite journal | author = Gewin V | year = 2006 | title = Beyond Neutrality—Ecology Finds Its Niche | journal = PLOS Biol | volume = 4 | issue = 8| pages = 1306–1310 | doi = 10.1371/journal.pbio.0040278 | pmid=16895443 | pmc=1543692 | doi-access = free }}</ref>]]

===Niche theory===
{{Main|Niche models}}

===Metapopulations===
{{See also|Metapopulation|Patch dynamics}}

Spatial analysis of ecological systems often reveals that assumptions that are valid for spatially homogenous populations – and indeed, intuitive – may no longer be valid when migratory subpopulations moving from one patch to another are considered.<ref>[[Ilkka Hanski|Hanski I]] (1999) [https://books.google.com/books?id=jsk4Nt_8X8sC&dq=Metapopulation+ecology&pg=PA1 ''Metapopulation ecology''] Oxford University Press. {{ISBN|978-0-19-854065-6}}.</ref> In a simple one-species formulation, a subpopulation may occupy a patch, move from one patch to another empty patch, or die out leaving an empty patch behind. In such a case, the proportion of occupied patches may be represented as
:<math> \frac{dp}{dt} = m p (1-p) - e p </math>
where m is the rate of [[Colonisation (biology)|colonization]], and e is the rate of [[extinction]].<ref>{{cite journal |vauthors=Hanski I, Gilpin M | year = 1991 | title = Metapopulation dynamics: brief history and conceptual domain | url = http://www.helsinki.fi/~ihanski/Articles/Biol_J_Linn_Soc_42.pdf | journal = Biological Journal of the Linnean Society | volume = 42 | issue = 1–2| pages = 3–16 | doi = 10.1111/j.1095-8312.1991.tb00548.x }}</ref> In this model, if e < m, the steady state value of p is 1 – (e/m) while in the other case, all the patches will eventually be left empty. This model may be made more complex by addition of another species in several different ways, including but not limited to game theoretic approaches, predator–prey interactions, etc. We will consider here an extension of the previous one-species system for simplicity. Let us denote the proportion of patches occupied by the first population as p<sub>1</sub>, and that by the second as p<sub>2</sub>. Then,
:<math> \frac{dp_1}{dt} = m_1 p_1 (1 - p_1) - e p_1 </math>
:<math> \frac{dp_2}{dt} = m_2 p_2 (1 - p_1 - p_2) - e p_2 - m p_1 p_2 </math>
In this case, if e is too high, p<sub>1</sub> and p<sub>2</sub> will be zero at steady state. However, when the rate of extinction is moderate, p<sub>1</sub> and p<sub>2</sub> can stably coexist. The steady state value of p<sub>2</sub> is given by
:<math> p^*_2 = \frac{e}{m_1} - \frac{m_1}{m_2} </math>
(p*<sub>1</sub> may be inferred by symmetry).
If e is zero, the dynamics of the system favor the species that is better at colonizing (i.e. has the higher m value). This leads to a very important result in theoretical ecology known as the [[Intermediate Disturbance Hypothesis]], where the [[biodiversity]] (the number of species that coexist in the population) is maximized when the disturbance (of which e is a proxy here) is not too high or too low, but at intermediate levels.<ref>Cox CB and Moore PD (2010) [https://books.google.com/books?id=GP5HeCwkV2IC&dq=%22Intermediate+Disturbance+Hypothesis%22&pg=PA146 ''Biogeography: An Ecological and Evolutionary Approach''] John Wiley and Sons, page 146. {{ISBN|978-0-470-63794-4}}.</ref>

The form of the differential equations used in this simplistic modelling approach can be modified. For example:
# Colonization may be dependent on p linearly (m*(1-p)) as opposed to the non-linear m*p*(1-p) regime described above. This mode of replication of a species is called the “rain of propagules”, where there is an abundance of new individuals entering the population at every generation. In such a scenario, the steady state where the population is zero is usually unstable.<ref name=Vandermeer>Vandermeer JH and Goldberg DE (2003) [https://books.google.com/books?id=DXCBTyB2GKIC&dq=%22rain+of+propagules%22+%22rescue+effect%22&pg=PA175 ''Population ecology: first principles''] Princeton University Press, page 175–176. {{ISBN|978-0-691-11441-5}}.</ref>
# Extinction may depend non-linearly on p (e*p*(1-p)) as opposed to the linear (e*p) regime described above. This is referred to as the “[[rescue effect]]” and it is again harder to drive a population extinct under this regime.<ref name=Vandermeer />

The model can also be extended to combinations of the four possible linear or non-linear dependencies of colonization and extinction on p are described in more detail in.<ref>
{{cite journal
 |author = Ilkka Hanski
 |year = 1982
 |title = Dynamics of Regional Distribution: The Core and Satellite Species Hypothesis
 |journal = Oikos
 |volume = 38
 |issue = 2
 |pages = 210–221
 |jstor= 3544021
 |doi=10.2307/3544021
|bibcode = 1982Oikos..38..210H
 }}
</ref>

==Ecosystem ecology==
{{See also|Ecosystem models}}

Introducing new elements, whether [[Biotic component|biotic]] or [[abiotic]], into [[ecosystem]]s can be disruptive. In some cases, it leads to [[ecological collapse]], [[trophic cascade]]s and the death of many species within the ecosystem. The abstract notion of [[ecological health]] attempts to measure the robustness and recovery capacity for an ecosystem; i.e. how far the ecosystem is away from its steady state. Often, however, ecosystems rebound from a disruptive agent. The difference between collapse or rebound depends on the [[toxicity]] of the introduced element and the [[resilience (ecology)|resiliency]] of the original ecosystem.

If ecosystems are governed primarily by [[stochastic]] processes, through which its subsequent state would be determined by both predictable and random actions, they may be more resilient to sudden change than each species individually. In the absence of a [[balance of nature]], the [[species composition]] of ecosystems would undergo shifts that would depend on the nature of the change, but entire ecological collapse would probably be infrequent events. In 1997, [[Robert Ulanowicz]] used [[information theory]] tools to describe the structure of ecosystems, emphasizing [[mutual information]] (correlations) in studied systems. Drawing on this methodology and prior observations of complex ecosystems, Ulanowicz depicts approaches to determining the stress levels on ecosystems and predicting system reactions to defined types of alteration in their settings (such as increased or reduced energy flow), and [[eutrophication]].<ref>Robert Ulanowicz (). ''Ecology, the Ascendant Perspective''. Columbia Univ. Press. {{ISBN|0-231-10828-1}}.</ref>

[[Ecopath]] is a free ecosystem modelling software suite, initially developed by [[NOAA]], and widely used in fisheries management as a tool for modelling and visualising the complex relationships that exist in real world marine ecosystems.

===Food webs===
[[Food web]]s provide a framework within which a complex network of predator–prey interactions can be organised. A food web model is a network of [[food chain]]s. Each food chain starts with a [[primary producer]] or [[autotroph]], an organism, such as a plant, which is able to manufacture its own food. Next in the chain is an organism that feeds on the primary producer, and the chain continues in this way as a string of successive predators. The organisms in each chain are grouped into [[trophic level]]s, based on how many links they are removed from the primary producers. The length of the chain, or trophic level, is a measure of the number of species encountered as energy or nutrients move from plants to top predators.<ref name="Post93">{{cite journal|last=Post|first=D. M.|title= The long and short of food-chain length|year=1993|journal = Trends in Ecology and Evolution|volume=17|issue=6| pages=269–277|doi=10.1016/S0169-5347(02)02455-2}}</ref> [[Food energy]] flows from one organism to the next and to the next and so on, with some energy being lost at each level.  At a given trophic level there may be one species or a group of species with the same predators and prey.<ref>{{cite book | author= Jerry Bobrow, Ph.D.|author2=Stephen Fisher | title= CliffsNotes CSET: Multiple Subjects| year= 2009| edition= 2nd| publisher= John Wiley and Sons| page= 283| url= https://books.google.com/books?id=BAaYNjlrJDcC&q=%22presumed+to+share+both+predators+and+prey%22&pg=PR1| isbn= 978-0-470-45546-3}}</ref>

In 1927, [[Charles Sutherland Elton|Charles Elton]] published an influential synthesis on the use of food webs, which resulted in them becoming a central concept in ecology.<ref>Elton CS (1927) Animal Ecology. Republished 2001. University of Chicago Press.</ref> In 1966, interest in food webs increased after [[Robert T. Paine (zoologist)|Robert Paine's]] experimental and descriptive study of intertidal shores, suggesting that food web complexity was key to maintaining species diversity and ecological stability.<ref>{{cite journal | author = Paine RT | year = 1966 | title = Food web complexity and species diversity | journal = The American Naturalist | volume = 100 | issue = 910| pages = 65–75 | doi = 10.1086/282400 | s2cid = 85265656 }}</ref> Many theoretical ecologists, including [[Robert May, Baron May of Oxford|Sir Robert May]] and [[Stuart Pimm]], were prompted by this discovery and others to examine the mathematical properties of food webs. According to their analyses, complex food webs should be less stable than simple food webs.<ref name=May />{{rp|75–77}}<ref name=Pimm />{{rp|64}} The apparent paradox between the complexity of food webs observed in nature and the mathematical fragility of food web models is currently an area of intensive study and debate. The paradox may be due partially to conceptual differences between persistence of a food web and equilibrial [[Ecological stability|stability]] of a food web.<ref name=May>May RM (2001) [https://books.google.com/books?id=BDA5-ipCLt4C&dq=%22Stability+and+Complexity+in+Model+Ecosystems%22&pg=PR7 ''Stability and Complexity in Model Ecosystems''] Princeton University Press, reprint of 1973 edition with new foreword. {{ISBN|978-0-691-08861-7}}.</ref><ref name=Pimm>Pimm SL (2002) [https://books.google.com/books?id=tjHOtK4amfQC&q=Pimm+%22Food+Webs%22 ''Food Webs''] University of Chicago Press, reprint of 1982 edition with new foreword. {{ISBN|978-0-226-66832-1}}.</ref>

===Systems ecology===
[[Systems ecology]] can be seen as an application of [[general systems theory]] to ecology. It takes a [[holistic]] and interdisciplinary approach to the study of ecological systems, and particularly ecosystems. Systems ecology is especially concerned with the way the functioning of ecosystems can be influenced by human interventions. Like other fields in theoretical ecology, it uses and extends concepts from [[thermodynamics]] and develops other macroscopic descriptions of complex systems. It also takes account of the [[Ecological energetics|energy flows]] through the different [[trophic level]]s in the ecological networks. Systems ecology also considers the external influence of [[ecological economics]], which usually is not otherwise considered in ecosystem ecology.<ref>R.L. Kitching, ''Systems ecology'', University of Queensland Press, 1983, p.9.</ref> For the most part, systems ecology is a subfield of ecosystem ecology.

==Ecophysiology==
{{main|Ecophysiology}}

This is the study of how "the environment, both physical and biological, interacts with the physiology of an organism. It includes the effects of climate and nutrients on physiological processes in both plants and animals, and has a particular focus on how physiological processes scale with organism size".<ref>[https://www.nature.com/subjects/ecophysiology Ecophysiology], ''Nature''. Date accessed: 9 August 2017.</ref><ref>[[Robert Henry Peters|Peters, R. H.]] (1986) [https://books.google.com/books?id=OYVxiZgTXWsC&dq=%22The+Ecological+Implications+of+Body+Size%22&pg=PR9 ''The Ecological Implications of Body Size''] Cambridge University Press. {{ISBN|9780521288866}}</ref>

==Behavioral ecology==
{{main|Behavioral ecology}}

===Swarm behaviour===
[[File:A wedge of starlings - geograph.org.uk - 1069366.jpg|thumb|right|Flocks of birds can abruptly change their direction in unison, and then, just as suddenly, make a unanimous group decision [[Bird landings|to land]].<ref name="Bhattacharya">Bhattacharya K and Vicsek T (2010) [https://arxiv.org/abs/1007.4453 "Collective decision making in cohesive flocks"]</ref>]]
{{See also|Swarm models}}

[[Swarm behaviour]] is a [[Collective animal behaviour|collective behaviour]] exhibited by animals of similar size which aggregate together, perhaps milling about the same spot or perhaps [[Animal migration|migrating]] in some direction. Swarm behaviour is commonly exhibited by insects, but it also occurs in the [[flocking (behavior)|flocking]] of birds, the [[shoaling and schooling|schooling]] of fish and the [[herd behaviour]] of quadrupeds. It is a complex [[emergent organization|emergent]] behaviour that occurs when [[Agent-based model in biology|individual agents]] follow simple behavioral rules.

Recently, a number of mathematical models have been discovered which explain many aspects of the emergent behaviour. Swarm algorithms follow a [[Lagrangian mechanics|Lagrangian]] approach or an [[Euler equations (fluid dynamics)|Eulerian]] approach.<ref name="Li et al">{{cite journal | author = Li YX | year = 2007 | title = Minimal mechanisms for school formation in self-propelled particles | url = http://www.iam.ubc.ca/~lukeman/fish_school_f.pdf | archive-url = https://web.archive.org/web/20111001032730/http://www.iam.ubc.ca/~lukeman/fish_school_f.pdf | url-status = dead | archive-date = 2011-10-01 | journal = Physica D: Nonlinear Phenomena | volume = 237 | issue = 5 | pages = 699–720 | doi = 10.1016/j.physd.2007.10.009 | author2 = Lukeman R | author3 = Edelstein-Keshet L | bibcode = 2008PhyD..237..699L }}</ref> The Eulerian approach views the swarm as a [[Field (physics)|field]], working with the density of the swarm and deriving mean field properties. It is a hydrodynamic approach, and can be useful for modelling the overall dynamics of large swarms.<ref>Toner J and Tu Y (1995) "Long-range order in a two-dimensional xy model: how birds fly together" ''Physical Revue Letters,'' '''75''' (23)(1995), 4326–4329.</ref><ref>{{cite journal |vauthors=Topaz C, Bertozzi A | year = 2004 | title = Swarming patterns in a two-dimensional kinematic model for biological groups | journal = SIAM J Appl Math | volume = 65 | issue = 1| pages = 152–174 | doi = 10.1137/S0036139903437424 | citeseerx = 10.1.1.88.3071 | bibcode = 2004APS..MAR.t9004T | s2cid = 18468679 }}</ref><ref>{{cite journal |vauthors=Topaz C, Bertozzi A, Lewis M | year = 2006 | title = A nonlocal continuum model for biological aggregation | journal = Bull Math Biol | volume = 68 | issue = 7| pages = 1601–1623 | doi = 10.1007/s11538-006-9088-6 | arxiv = q-bio/0504001 | pmid=16858662| s2cid = 14750061 }}</ref> However, most models work with the Lagrangian approach, which is an [[agent-based model]] following the individual agents (points or particles) that make up the swarm. Individual particle models can follow information on heading and spacing that is lost in the Eulerian approach.<ref name="Li et al"/><ref>{{cite book | last1 = Carrillo | first1 = J | last2 = Fornasier | first2 = M | last3 = Toscani | first3 = G | title = Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences | chapter = Particle, kinetic, and hydrodynamic models of swarming | series = Modeling and Simulation in Science, Engineering and Technology | year = 2010 | url = http://mate.unipv.it/~toscani/publi/swarming.pdf | volume = 3 | pages = 297–336 | doi = 10.1007/978-0-8176-4946-3_12 | isbn = 978-0-8176-4945-6 | citeseerx = 10.1.1.193.5047 }}</ref> Examples include [[ant colony optimization]], [[self-propelled particles]] and [[particle swarm optimization]].

On cellular levels, individual organisms also demonstrated swarm behavior. [[Decentralised system|Decentralized systems]] are where individuals act based on their own decisions without overarching guidance. Studies have shown that individual ''[[Trichoplax adhaerens]]'' behave like [[self-propelled particles]] (SPPs) and collectively display phase transition from ordered movement to disordered movements.<ref>{{Cite journal |last1=Davidescu |first1=Mircea R. |last2=Romanczuk |first2=Pawel |last3=Gregor |first3=Thomas |last4=Couzin |first4=Iain D. |date=2023-03-14 |title=Growth produces coordination trade-offs in Trichoplax adhaerens, an animal lacking a central nervous system |journal=Proceedings of the National Academy of Sciences |language=en |volume=120 |issue=11 |pages=e2206163120 |doi=10.1073/pnas.2206163120 |issn=0027-8424 |pmc=10089153 |pmid=36897970|bibcode=2023PNAS..12006163D }}</ref> Previously, it was thought that the surface-to-volume ratio was what limited the animal size in the evolutionary game. Considering the collective behaviour of the individuals, it was suggested that order is another limiting factor. [[Central nervous system]]s were indicated to be vital for large multicellular animals in the evolutionary pathway.

=== Synchronization ===
The coexistence of the [[synchronization]] and asynchronization in the flashings in the system composed of multiple fireflies could be characterized by the chimera states. Synchronization could spontaneously occur.<ref>{{Cite journal |last1=Sarfati |first1=Raphaël |last2=Peleg |first2=Orit |date=2022-11-18 |title=Chimera states among synchronous fireflies |journal=Science Advances |language=en |volume=8 |issue=46 |pages=eadd6690 |doi=10.1126/sciadv.add6690 |issn=2375-2548 |pmc=9668303 |pmid=36383660|bibcode=2022SciA....8D6690S }}</ref> The [[agent-based model]] has been useful in describing this unique phenomenon. The flashings of individual fireflies could be viewed as oscillators and the global coupling models were similar to the ones used in [[condensed matter physics]].

==Evolutionary ecology==
{{Main|Evolutionary ecology}}

The British biologist [[Alfred Russel Wallace]] is best known for independently proposing a theory of [[evolution]] due to [[natural selection]] that prompted [[Charles Darwin]] to publish his own theory. In his famous 1858 paper, Wallace proposed natural selection as a kind of feedback mechanism which keeps species and varieties adapted to their environment.<ref name="Unfinished Business"/>

<blockquote>''The action of this principle is exactly like that of the [[centrifugal governor]] of the steam engine, which checks and corrects any irregularities almost before they become evident; and in like manner no unbalanced deficiency in the animal kingdom can ever reach any conspicuous magnitude, because it would make itself felt at the very first step, by rendering existence difficult and extinction almost sure soon to follow.''<ref name="tendency">{{cite web|last=Wallace|first=Alfred|title=On the Tendency of Varieties to Depart Indefinitely From the Original Type|url=http://www.wku.edu/~smithch/wallace/S043.htm|publisher=The Alfred Russel Wallace Page hosted by [[Western Kentucky University]]|access-date=2007-04-22}}</ref></blockquote>

The [[cybernetics|cybernetician]] and anthropologist [[Gregory Bateson]] observed in the 1970s that, though writing it only as an example, Wallace had "probably said the most powerful thing that’d been said in the 19th Century".<ref>{{cite web|last=Brand|first=Stewart|title=For God's Sake, Margaret|url=http://www.oikos.org/forgod.htm|publisher=CoEvolutionary Quarterly, June 1976|access-date=2007-04-04}}</ref> Subsequently, the connection between natural selection and [[systems theory]] has become an area of active research.<ref name="Unfinished Business">{{cite web|last=Smith|first=Charles H.|title=Wallace's Unfinished Business|url=http://www.wku.edu/~smithch/essays/UNFIN.htm|publisher=Complexity (publisher Wiley Periodicals, Inc.) Volume 10, No 2, 2004|access-date=2007-05-11}}</ref>

==Other theories==
In contrast to previous ecological theories which considered [[floods]] to be catastrophic events, the river [[flood pulse concept]] argues that the annual flood pulse is the most important aspect and the most biologically productive feature of a [[lotic ecosystem|river's ecosystem]].<ref name=thorp>Thorp, J. H., & Delong, M. D. (1994). The Riverine Productivity Model: An Heuristic View of Carbon Sources and Organic Processing in Large River Ecosystems. Oikos, 305-308</ref><ref>Benke, A. C., Chaubey, I., Ward, G. M., & Dunn, E. L. (2000). Flood Pulse Dynamics of an Unregulated River Floodplain in the Southeastern U.S. Coastal Plain. Ecology, 2730-2741.</ref>

==History==
{{See also|History of ecology}}

Theoretical ecology draws on pioneering work done by [[G. Evelyn Hutchinson]] and his students.  Brothers [[Howard T. Odum|H.T. Odum]] and [[Eugene Odum|E.P. Odum]] are generally recognised as the founders of modern theoretical ecology. [[Robert MacArthur]] brought theory to [[community ecology]]. [[Daniel Simberloff]] was the student of [[E.O. Wilson]], with whom MacArthur collaborated on ''[[The Theory of Island Biogeography]]'', a seminal work in the development of theoretical ecology.<ref name=Cuddington>Cuddington K and Beisner BE (2005) [https://books.google.com/books?id=5J0MhQaiP-sC&q=intitle:Ecological+intitle:paradigms+intitle:lost ''Ecological paradigms lost: routes of theory change''] Academic Press. {{ISBN|978-0-12-088459-9}}.</ref>

Simberloff added statistical rigour to [[experimental ecology]] and was a key figure in the [[SLOSS debate]], about whether it is preferable to protect a single large or several small reserves.<ref name=Soule>{{cite journal |vauthors=Soulé ME, Simberloff D | year = 1986 | title = What do genetics and ecology tell us about the design of nature reserves? | url = http://deepblue.lib.umich.edu/bitstream/2027.42/26318/1/0000405.pdf | journal = Biological Conservation | volume = 35 | issue = 1| pages = 19–40 | doi = 10.1016/0006-3207(86)90025-X | hdl = 2027.42/26318 | hdl-access = free }}</ref> This resulted in the supporters of [[Jared Diamond]]'s community assembly rules defending their ideas through Neutral Model Analysis.<ref name=Soule />  Simberloff also played a key role in the (still ongoing) debate on the utility of corridors for connecting isolated reserves.

[[Stephen P. Hubbell]] and [[Michael Rosenzweig]] combined theoretical and practical elements into works that extended MacArthur and Wilson's Island Biogeography Theory - Hubbell with his [[Unified neutral theory of biodiversity|Unified Neutral Theory of Biodiversity and Biogeography]] and Rosenzweig with his Species Diversity in Space and Time.

==Theoretical and mathematical ecologists==
A tentative distinction can be made between mathematical ecologists, ecologists who apply mathematics to ecological problems, and mathematicians who develop the mathematics itself that arises out of ecological problems.

Some notable theoretical ecologists can be found in these categories:

* [[:Category:Mathematical ecologists]]
* [[:Category:Theoretical biologists]]

==Journals==
{{Div col|colwidth=24em}}
* ''[[The American Naturalist]]''
* ''[[Journal of Mathematical Biology]]''
* ''[[Journal of Theoretical Biology]]''
* ''Theoretical Ecology''
* ''Theoretical Population Biology''
* ''Ecological Modelling''
{{div col end}}

==See also==
{{Div col|colwidth=24em}}
* [[Butterfly effect]]
* [[Complex system biology]]
* [[Ecological systems theory]]
* [[Ecosystem model]]
* [[Integrodifference equation]] – widely used to model the [[biological dispersal|dispersal]] and growth of populations
* [[Limiting similarity]]
* [[Mathematical biology]]
* [[Population dynamics]]
* [[Population modeling]]
* [[Quantitative ecology]]
* [[Taylor's law]]
* [[Theoretical biology]]
* [[Early-bird effect]]
{{div col end}}

==References==
{{Reflist}}

==Further reading==
{{refbegin}}
* The classic text is ''Theoretical Ecology: Principles and Applications'', by [[Angela McLean]] and [[Robert May, Baron May of Oxford|Robert May]]. The 2007 edition is published by the Oxford University Press. {{ISBN|978-0-19-920998-9}}.
{{refend}}
----
{{refbegin|2}}
* Bolker BM (2008) [https://books.google.com/books?id=yULf2kZSfeMC&q=intitle:Ecological+intitle:models+intitle:and+intitle:data+intitle:in+intitle:R ''Ecological Models and Data in R''] Princeton University Press. {{ISBN|978-0-691-12522-0}}.
* Case TJ (2000) [https://books.google.com/books?id=eZvYRBPWmkoC&q=%22Theoretical+ecology%22 ''An illustrated guide to theoretical ecology''] Oxford University Press. {{ISBN|978-0-19-508512-9}}.
* Caswell H (2000) ''Matrix Population Models: Construction, Analysis, and Interpretation'', Sinauer, 2nd Ed. {{ISBN|978-0-87893-096-8}}.
* Edelstein-Keshet L (2005) [https://books.google.com/books?id=pp9pQgAACAAJ&q=intitle:Mathematical+intitle:Models+intitle:in+intitle:Biology ''Mathematical Models in Biology''] Society for Industrial and Applied Mathematics. {{ISBN|978-0-89871-554-5}}.
* Gotelli NJ (2008) [https://books.google.com/books?id=q7_0LwAACAAJ&q=intitle:A+intitle:Primer+intitle:of+intitle:Ecology ''A Primer of Ecology''] Sinauer Associates, 4th Ed. {{ISBN|978-0-87893-318-1}}.
* Gotelli NJ & A Ellison (2005) [https://books.google.com/books?id=vaZ1QgAACAAJ&q=intitle:A+intitle:Primer+intitle:Of+intitle:Ecological+intitle:Statistics ''A Primer Of Ecological Statistics''] Sinauer Associates Publishers. {{ISBN|978-0-87893-269-6}}.
* Hastings A (1996) [https://books.google.com/books?id=1cFozFXjjlAC&q=intitle:Population+intitle:Biology+intitle:Concepts+intitle:and+intitle:Models ''Population Biology: Concepts and Models''] Springer. {{ISBN|978-0-387-94853-9}}.
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* {{cite journal | author = Lawton JH | year = 1999 | title = Are there general laws in ecology? | url = http://lamar.colostate.edu/~aknapp/ey505/Lawton%20Laws%20in%20Ecology%201999.pdf | archive-url = https://web.archive.org/web/20100611161533/http://lamar.colostate.edu/%7Eaknapp/ey505/Lawton%20Laws%20in%20Ecology%201999.pdf | url-status = dead | archive-date = 2010-06-11 | journal = Oikos | volume = 84 | issue = 2 | pages = 177–192 | doi = 10.2307/3546712 | jstor = 3546712 | bibcode = 1999Oikos..84..177L | citeseerx = 10.1.1.331.1173 }}
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* Murray JD (2003) [https://books.google.com/books?id=2d-RLuD_MX8C&q=intitle:Mathematical+intitle:Biology+Murray ''Mathematical Biology, Volume 2''] Springer, 3rd Ed. {{ISBN|978-0-387-95228-4}}.
* Pastor J (2008) [https://books.google.com/books?id=PipFAQAAIAAJ&q=intitle:Mathematical+intitle:Ecology+intitle:of+intitle:Populations+intitle:and+intitle:Ecosystems ''Mathematical Ecology of Populations and Ecosystems''] Wiley-Blackwell. {{ISBN|978-1-4051-8811-1}}.
* Roughgarden J (1998) [https://books.google.com/books?id=PmHwAAAAMAAJ&q=intitle:Primer+intitle:of+intitle:Ecological+intitle:Theory ''Primer of Ecological Theory''] Prentice Hall. {{ISBN|978-0-13-442062-2}}.
* Ulanowicz R (1997) [https://books.google.com/books?id=jSKDw_zAMJUC ''Ecology: The Ascendant Perspective''] Columbia University Press.
{{refend}}

{{Branches of ecology}}
{{modelling ecosystems|expanded=other}}

{{DEFAULTSORT:Theoretical ecology}}
[[Category:Theoretical ecology| ]]
[[Category:Ecological theories|*]]
[[Category:Mathematical and theoretical biology|Ecology]]