Editing Lotka–Volterra equations

{{Short description|Equations modelling predator–prey cycles}}
{{About|the predator-prey equations|the competition equations|Competitive Lotka–Volterra equations}}

The '''Lotka–Volterra equations''', also known as the '''Lotka–Volterra predator–prey model''', are a pair of first-order [[nonlinear]] [[differential equation]]s, frequently used to describe the [[dynamical system|dynamics]] of [[Systems biology|biological systems]] in which two species interact, one as a [[predator]] and the other as prey. The [[population cycle|populations change through time]] according to the pair of equations:
<math display="block">\begin{align}
 \frac{dx}{dt} &= \alpha x - \beta x y, \\
 \frac{dy}{dt} &= - \gamma y + \delta x y ,
\end{align}</math>

where
*the [[Variable (mathematics)|variable]] {{mvar|x}} is the [[population density]] of prey (for example, the number of [[rabbit]]s per square kilometre);
*the variable {{mvar|y}} is the population density of some [[Predation|predator]] (for example, the number of [[fox]]es per square kilometre);
*<math>\tfrac{dy}{dt}</math> and <math>\tfrac{dx}{dt}</math> represent the instantaneous growth rates of the two populations;
*{{mvar|t}} represents time;
*The prey's [[parameter]]s, {{mvar|α}} and {{mvar|β}}, describe, respectively, the maximum prey [[per capita]] growth rate, and the effect of the presence of predators on the prey death rate. 
*The predator's parameters, {{mvar|γ}}, {{mvar|δ}}, respectively describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate. 
*All parameters are positive and real.

The solution of the differential equations is [[deterministic system|deterministic]] and [[Continuous function|continuous]]. This, in turn, implies that the generations of both the predator and prey are continually overlapping.<ref>{{cite book|last1=Cooke|first1=D.|last2=Hiorns|first2=R. W.|display-authors=etal|title=The Mathematical Theory of the Dynamics of Biological Populations|volume=II|publisher=Academic Press|year=1981}}</ref>

The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known [[Kolmogorov equations]]),<ref>{{cite book |last=Freedman |first=H. I. |title=Deterministic Mathematical Models in Population Ecology |publisher=[[Marcel Dekker]] |year=1980}}</ref><ref>{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chavez |first2=C. |title=Mathematical Models in Population Biology and Epidemiology |publisher=[[Springer-Verlag]] |year=2000}}</ref><ref name="scholarpedia">{{cite journal | last=Hoppensteadt |first=F. |title=Predator-prey model |journal=[[Scholarpedia]] |volume=1 |issue=10 |page=1563 | year=2006| doi=10.4249/scholarpedia.1563 |bibcode=2006SchpJ...1.1563H |doi-access=free }}</ref> which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, [[Competition (biology)|competition]], disease, and [[mutualism (biology)|mutualism]].

== Biological interpretation and model assumptions ==

The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this [[exponential growth]] is represented in the equation above by the term {{math|''αx''}}. The rate of predation on the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by {{math|''βxy''}}. If either {{mvar|x}} or {{mvar|y}} is zero, then there can be no predation. With these two terms the prey equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon.

The term {{math|''δxy''}} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term {{math|''γy''}} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey. Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate.

The Lotka–Volterra predator-prey model makes a number of assumptions about the environment and biology of the predator and prey populations:<ref>{{Cite web |last=Beals |first=M. |last2=Gross |first2=L. |last3=Townsend |first3=C.R. |date=1999 |title=Predator-Prey Dynamics |url=http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html |url-status=dead |archive-url=https://web.archive.org/web/20121215031911/http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html |archive-date=2012-12-15 |access-date=2018-01-09 |website=www.tiem.utk.edu}}</ref>
# The prey population finds ample food at all times.
# The food supply of the predator population depends entirely on the size of the prey population.
# The rate of change of population is proportional to its size.
# During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential.
# Predators have limitless appetite.
# Both populations can be described by a single variable. This amounts to assuming that the populations do not have a spatial or age distribution that contributes to the dynamics.

== Biological relevance of the model ==
[[File:Milliers fourrures vendues en environ 90 ans odum 1953 en.jpg|upright=1.5|thumb|Numbers of [[Lepus americanus|snowshoe hare]] (yellow, background) and [[Canada lynx]] (black line, foreground) furs sold to the [[Hudson's Bay Company]]. Canada lynxes eat snowshoe hares.]]

None of the assumptions above are likely to hold for natural populations. Nevertheless, the Lotka–Volterra model shows two important properties of predator and prey populations and these properties often extend to variants of the model in which these assumptions are relaxed:

Firstly, the dynamics of predator and prey populations have a tendency to oscillate. Fluctuating numbers of predators and prey have been observed in natural populations, such as the [[lynx]] and [[snowshoe hare]] data of the [[Hudson's Bay Company]]<ref>{{cite journal|last=Gilpin|first=M. E.| year=1973|title=Do hares eat lynx?|journal=American Naturalist|issue=957|pages=727–730|doi=10.1086/282870| volume=107| s2cid=84794121}}</ref> and the moose and wolf populations in [[Isle Royale National Park]].<ref>{{cite journal|last1=Jost| first1=C.|last2=Devulder|first2=G.|last3=Vucetich|first3=J.A.| last4=Peterson|first4=R.|last5=Arditi|first5=R.| doi=10.1111/j.1365-2656.2005.00977.x|title=The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose|journal=J. Anim. Ecol.|volume=74|issue=5|pages=809–816| year=2005}}</ref>

Secondly, the population equilibrium of this model has the property that the prey equilibrium density (given by <math> x = \gamma/\delta </math>) depends on the predator's parameters, and the predator equilibrium density (given by <math> y =  \alpha/\beta </math>) on the prey's parameters. This has as a consequence that an increase in, for instance, the prey growth rate, <math> \alpha</math>, leads to an increase in the predator equilibrium density, but not the prey equilibrium density. Making the environment better for the prey benefits the predator, not the prey (this is related to the [[paradox of the pesticides]] and to the [[paradox of enrichment]]). A demonstration of this phenomenon is provided by the increased percentage of predatory fish caught had increased during the years of [[World War I]] (1914–18), when prey growth rate was increased due to a reduced fishing effort.

A further example is provided by the experimental [[iron fertilization]] of the ocean. In several [[iron fertilization#Experiments|experiments]] large amounts of iron salts were dissolved in the ocean. The expectation was that iron, which is a limiting nutrient for phytoplankton, would boost growth of phytoplankton and that it would sequester carbon dioxide from the atmosphere. The addition of iron typically leads to a short bloom in phytoplankton, which is quickly consumed by other organisms (such as small fish or [[zooplankton]]) and limits the effect of enrichment mainly to increased predator density, which in turn limits the [[carbon sequestration]]. This is as predicted by the equilibrium population densities of the Lotka–Volterra predator-prey model, and is a feature that carries over to more elaborate models in which the restrictive assumptions of the simple model are relaxed.<ref>{{Cite journal | 
last1=Pan |first1=A. |last2=Pourziaei |first2=B.|last3=Huang|first3=H.|date=2015-06-03|title= Effect of Ocean Iron Fertilization on the Phytoplankton Biological Carbon Pump.|volume=3 |issue=1 |pages=52–64|doi=10.4208/aamm.10-m1023|journal=Advances in Applied Mathematics and Mechanics|s2cid=124606355 }}</ref>

== Applications to economics and marketing ==
The Lotka–Volterra model has additional applications to areas such as economics<ref>{{Cite book |last=Prasolov |first=Alexander V. |title=Some quantitative methods and models in economic theory |date=2016 |publisher=Nova Publishers |isbn=978-1-63484-937-1 |series=Economic issues, problems and perspectives |location=New York}}</ref> and marketing.<ref>{{Cite journal |last1=Hung |first1=Hui-Chih |last2=Chiu |first2=Yu-Chih |last3=Wu |first3=Muh-Cherng |date=2017 |title=A Modified Lotka–Volterra Model for Diffusion and Substitution of Multigeneration DRAM Processing Technologies |journal=Mathematical Problems in Engineering |volume=2017 |pages=1–12 |doi=10.1155/2017/3038203 |issn=1024-123X|doi-access=free }}</ref><ref>{{Cite book |last=Orbach |first=Yair |title=Forecasting the Dynamics of Market and Technology |publisher=Ariel University Press |year=2022 |isbn=978-965-7632-40-6 |location=Israel |pages=123–143 |language=English}}</ref> It can be used to describe the dynamics in a market with several competitors, complementary platforms and products, a sharing economy, and more. There are situations in which one of the competitors drives the other competitors out of the market and other situations in which the market reaches an equilibrium where each firm stabilizes on its market share. It is also possible to describe situations in which there are cyclical changes in the industry or chaotic situations with no equilibrium and changes are frequent and unpredictable.

In economics, the [[Phillips curve]], which shows the statistical relationship between unemployment and the rate of change in nominal wages, has been connected by the [[Goodwin model (economics)|Goodwin model]]. This model reinterprets the dynamics of the biological prey-predator interaction, as described by the Lotka-Volterra model, in economic terms. The way the two species interact in this model led Goodwin to draw parallels with the [[Marxian class theory|Marxian class]] [[Conflicts involving Critical Mass|conflict]]. The Kolmogorov generalization of the prey-predator model, along with further developments of the Goodwin model, has extended these ideas.<ref>{{Citation |last=Orlando |first=Giuseppe |title=Growth and Cycles as a Struggle: Lotka–Volterra, Goodwin and Phillips |date=2021 |work=Nonlinearities in Economics: An Interdisciplinary Approach to Economic Dynamics, Growth and Cycles |pages=191–208 |editor-last=Orlando |editor-first=Giuseppe |url=https://link.springer.com/chapter/10.1007/978-3-030-70982-2_14 |access-date=2025-04-20 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-70982-2_14 |isbn=978-3-030-70982-2 |last2=Sportelli |first2=Mario |editor2-last=Pisarchik |editor2-first=Alexander N. |editor3-last=Stoop |editor3-first=Ruedi}}</ref>

==History==

The Lotka–Volterra predator–prey [[mathematical model|model]] was initially proposed by [[Alfred J. Lotka]] in the theory of autocatalytic chemical reactions in 1910.<ref>{{cite journal|last=Lotka|first=A. J.|title=Contribution to the Theory of Periodic Reaction|journal=[[Journal of Physical Chemistry A|J. Phys. Chem.]]|volume=14|issue=3|pages=271–274|year=1910| doi=10.1021/j150111a004|url=https://zenodo.org/record/1428768}}</ref><ref name="Goelmany">{{cite book |last1=Goel |first1=N. S. |first2=S. C. |last2=Maitra |first3=E. W. |last3=Montroll |display-authors=1 |title=On the Volterra and Other Nonlinear Models of Interacting Populations |publisher=Academic Press|year=1971 |isbn=0-12-287450-1 }}</ref> This was effectively the [[Logistic function#In ecology: modeling population growth|logistic equation]],<ref>{{cite journal|last=Berryman|first=A. A.| url=http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|title=The Origins and Evolution of Predator-Prey Theory|journal=[[Ecology (journal)|Ecology]]|volume=73|issue=5|pages=1530–1535|year=1992|url-status=dead|archive-url=https://web.archive.org/web/20100531204042/http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|archive-date=2010-05-31|doi=10.2307/1940005|jstor=1940005}}</ref> originally derived by [[Pierre François Verhulst]].<ref>{{cite journal|last=Verhulst|first=P. H.|url=https://books.google.com/books?id=8GsEAAAAYAAJ|title=Notice sur la loi que la population poursuit dans son accroissement|journal=Corresp. Mathématique et Physique|volume=10|pages=113–121|year=1838}}</ref> In 1920 Lotka extended the model, via [[Andrey Kolmogorov]], to "organic systems" using a plant species and a herbivorous animal species as an example<ref>{{cite journal|last=Lotka|first=A. J.|pmc=1084562|title=Analytical Note on Certain Rhythmic Relations in Organic Systems|journal=[[Proc. Natl. Acad. Sci. U.S.A.]]|volume=6|issue=7|pages=410–415|year=1920|doi=10.1073/pnas.6.7.410|pmid=16576509|bibcode=1920PNAS....6..410L|doi-access=free}}</ref> and in 1925 he used the equations to analyse predator–prey interactions in his book on [[biomathematics]].<ref>{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|publisher=[[Williams and Wilkins]]|year=1925}}</ref> The same set of equations was published in 1926 by [[Vito Volterra]], a mathematician and physicist, who had become interested in [[mathematical biology]].<ref name="Goelmany"/><ref>{{cite journal|last=Volterra|first=V.|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|year=1926}}</ref><ref>{{cite book|last=Volterra|first=V.|chapter=Variations and fluctuations of the number of individuals in animal species living together|title=Animal Ecology|editor-last=Chapman|editor-first=R. N.|publisher=[[McGraw–Hill]]|year=1931}}</ref> Volterra's enquiry was inspired through his interactions with the marine biologist [[Umberto D'Ancona]], who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the [[Adriatic Sea]] and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years and, as prey fish the preferred catch, one would intuitively expect this to increase of prey fish percentage. Volterra developed his model to explain D'Ancona's observation and did this independently from Alfred Lotka. He did credit Lotka's earlier work in his publication, after which the model has become known as the "Lotka-Volterra model".<ref>{{cite book|last=Kingsland|first=S.| title=Modeling Nature: Episodes in the History of Population Ecology|publisher=University of Chicago Press|year=1995| isbn=978-0-226-43728-6}}</ref>

The model was later extended to include density-dependent prey growth and a [[functional response]] of the form developed by [[C. S. Holling]]; a model that has become known as the Rosenzweig–MacArthur model.<ref>{{cite journal|last1=Rosenzweig|first1=M. L.|last2=MacArthur|first2=R.H.|year=1963|title=Graphical representation and stability conditions of predator-prey interactions|journal=American Naturalist|issue=895|pages=209–223|doi=10.1086/282272|volume=97|s2cid=84883526}}</ref> Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey.

In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or [[Arditi–Ginzburg equations|Arditi–Ginzburg model]].<ref>{{cite journal|last1=Arditi|first1=R.|last2=Ginzburg|first2=L. R.|year=1989|url=http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf|title=Coupling in predator-prey dynamics: ratio dependence|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 validity of prey- or ratio-dependent models has been much debated.<ref>{{cite journal|last1=Abrams|first1=P. A.|last2=Ginzburg|first2=L. R.| year=2000|title=The nature of predation: prey dependent, ratio dependent or neither?|journal=Trends in Ecology & Evolution| volume=15| issue=8|pages=337–341|doi=10.1016/s0169-5347(00)01908-x|pmid=10884706}}</ref>

The Lotka–Volterra equations have a long history of use in [[Economics|economic theory]]; their initial application is commonly credited to [[Richard M. Goodwin|Richard Goodwin]] in 1965<ref>{{cite journal|last=Gandolfo|first=G.|author-link=Giancarlo Gandolfo|title=Giuseppe Palomba and the Lotka–Volterra equations|journal=Rendiconti Lincei|volume=19|issue=4| pages=347–357| year=2008|doi=10.1007/s12210-008-0023-7|s2cid=140537163}}</ref> or 1967.<ref>{{cite book|last=Goodwin|first=R. M.|chapter=A Growth Cycle|title=Socialism, Capitalism and Economic Growth|chapter-url=https://archive.org/details/socialismcapital0000fein | chapter-url-access=registration|editor-last=Feinstein|editor-first=C. H.|publisher=[[Cambridge University Press]]|year=1967}}</ref><ref>{{cite journal|last1=Desai|first1=M.|last2=Ormerod|first2=P.|url=http://www.paulormerod.com/pdf/economicjournal1998.pdf |title=Richard Goodwin: A Short Appreciation|journal=[[The Economic Journal]]|volume=108 |issue=450 |pages=1431–1435|year=1998|doi=10.1111/1468-0297.00350|citeseerx=10.1.1.423.1705|access-date=2010-03-22|archive-url=https://web.archive.org/web/20110927154044/http://www.paulormerod.com/pdf/economicjournal1998.pdf|archive-date=2011-09-27|url-status=dead}}</ref>

==Solutions to the equations==
The equations have [[periodic function|periodic]] solutions.  These solutions do not have a simple expression in terms of the usual [[trigonometric function]]s, although they are quite tractable.<ref>{{cite journal|last1=Steiner|first1=Antonio| last2=Gander| first2=Martin Jakob|year=1999|title=Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich|journal=Il Volterriano|volume=7|pages=32–44|url=http://archive-ouverte.unige.ch/unige:6300/ATTACHMENT01}}</ref><ref>{{cite journal| last1=Evans|first1=C. M.|last2=Findley|first2=G. L.|title=A new transformation for the Lotka-Volterra problem| journal=Journal of Mathematical Chemistry|volume=25|pages=105–110|year=1999|doi=10.1023/A:1019172114300|s2cid=36980176}}</ref><ref>{{cite journal|
last1=Leconte|first1=M.|last2=Masson|first2=P.|last3=Qi|first3=L.|year=2022|title=Limit cycle oscillations, response time, and the time-dependent solution to the Lotka-Volterra predator-prey model| journal=Physics of Plasmas|volume=29|issue=2|pages=022302|arxiv=2110.11557|doi=10.1063/5.0076085|s2cid=239616189}}</ref>

If none of the non-negative parameters {{mvar|α}}, {{mvar|β}}, {{mvar|γ}}, {{mvar|δ}} vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in {{mvar|x}}, and the second one in {{mvar|y}}, the parameters ''β''/''α'' and ''δ''/''γ'' are absorbable in the normalizations of {{mvar|y}} and {{mvar|x}} respectively, and {{mvar|γ}} into the normalization of {{mvar|t}}, so that only {{math|''α''/''γ''}} remains arbitrary. It is the only parameter affecting the nature of the solutions.

[[Image:Lotka Volterra dynamics.svg|thumb|Prey and predator populations over time]]
A [[linearization]] of the equations yields a solution similar to [[simple harmonic motion]]<ref>{{cite book|last=Tong| first=H.| title=Threshold Models in Non-linear Time Series Analysis|publisher=Springer–Verlag|year=1983}}</ref> with the population of predators trailing that of prey by 90° in the cycle.
{{Further|Limit cycle}}

{{anchor|Atto-fox}}

===A simple example===
[[File:Lotka-Volterra model (1.1, 0.4, 0.4, 0.1).svg|alt=|thumb|400x400px|Population dynamics for rabbit and fox problem mentioned aside.]]
[[File:Predator prey dynamics.svg|alt=|thumb|300x300px|Phase-space plot for the predator prey problem for various initial conditions of the predator population.]]
Suppose there are two species of animals, a rabbit (prey) and a fox (predator). If the initial densities are 10 rabbits and 10 foxes per square kilometre, one can plot the progression of the two species over time; given the parameters that the growth and death rates of rabbits are 1.1 and 0.4 while that of foxes are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.

One may also plot solutions parametrically as [[orbit (dynamics)|orbit]]s in [[phase space]], without representing time, but with one axis representing the number of prey and the other axis representing the densities of predators for all times.

This corresponds to eliminating time from the two differential equations above to produce a single differential equation
:<math>\frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha}</math>
relating the variables ''x'' (predator) and ''y'' (prey). The solutions of this equation are closed curves. It is amenable to [[separation of variables]]: integrating
:<math>\frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0</math> 
yields the implicit relationship 
: <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y),</math>
where ''V'' is a constant quantity depending on the initial conditions and conserved on each curve.

An aside: These graphs illustrate a serious potential limitation in the application as a biological model: for this specific choice of parameters, in each cycle, the rabbit population is reduced to extremely low numbers, yet recovers (while the fox population remains sizeable at the lowest rabbit density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals might cause the rabbits to actually go extinct, and, by consequence, the foxes as well. This modelling problem has been called the "atto-fox problem", an [[atto-]]<nowiki>fox</nowiki> being a notional 10<sup>−18</sup> of a fox.<ref name="LobrySari2015">{{cite journal |last1=Lobry |first1=Claude |last2=Sari |first2=Tewfik |title=Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem |journal=Arima |date=2015 |volume=20 |pages=95–125 |url=http://arima.inria.fr/020/pdf/vol.20.pp.95-125.pdf}}</ref><ref>{{cite journal |last=Mollison |first=D. |url=http://www.ma.hw.ac.uk/~denis/epi/velocities.pdf |title=Dependence of epidemic and population velocities on basic parameters |journal=Math. Biosci. |volume=107 |issue=2 |pages=255–287 |year=1991 |doi=10.1016/0025-5564(91)90009-8 |pmid=1806118 }}</ref> A density of 10<sup>−18</sup> foxes per square kilometre equates to an average of approximately 5×10<sup>−10</sup> foxes on the surface of the earth, which in practical terms means that foxes are extinct.

===Hamiltonian structure of the system===
Since the quantity <math>V(x,y)</math> is conserved over time, it plays role of a Hamiltonian function of the system.<ref>{{cite journal|last1=Nutku|first1=I. |year=1990|title=Hamiltonian structure of the Lotka-Volterra equations|journal=Physics Letters A|volume=145|issue=1 |pages=27–28|url=https://www.sciencedirect.com/science/article/abs/pii/037596019090270X |doi=10.1016/0375-9601(90)90270-X|bibcode=1990PhLA..145...27N |hdl=11693/26204 |s2cid=121710034 |hdl-access=free}}</ref> To see this we can define [[Poisson bracket]] as follows	
<math> \{f(x,y), g(x,y)\} = -xy \left( \frac{\partial f}{\partial x}\frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right) </math>.  Then [[Hamilton's equations]] read
<math display="block">\begin{cases}
\dot{x} =  \{x, V\} = \alpha x - \beta x y, \\ 
\dot{y} =  \{y, V\} = \delta x y - \gamma y.
\end{cases}</math>
The variables <math>x</math> and <math>y</math> are not canonical, since <math> \{x, y\} = -xy \neq 1</math>. However, using transformations<ref>{{cite web |url=http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Bio-Mathematics-(APPLIED).pdf |title=Lotka-Volterra Dynamics - An introduction |last=Baigent |first=Steve |date=2010-03-02 }}</ref> <math> p = \ln (x)</math> and <math> q = \ln (y)</math> we came up to a canonical form of the [[Hamilton's equations]] featuring the Hamiltonian <math> H(q,p) = V(x(q,p),y(q,p)) = \delta e^p - \gamma p + \beta e^q - \alpha q </math>:
<math display="block">\begin{cases}
\dot{q} =  \frac{\partial H}{\partial p} = \delta e^p - \gamma, \\ 
\dot{p} =  -\frac{\partial H}{\partial q} = \alpha - \beta e^q.
\end{cases}</math>
The [[Poisson bracket]] for the canonical variables <math>(q,p)</math> now takes the standard form <math> \{F(q,p), G(q,p)\} = \left( \frac{\partial F}{\partial q}\frac{\partial G}{\partial p} - \frac{\partial F}{\partial p} \frac{\partial G}{\partial q} \right) </math>.

===Phase-space plot of a further example===
[[Image:Lotka-Volterra orbits 01.svg|alt=|left|thumb|300x300px]]Another example covers: 
{{math|1=''α'' = 2/3}}, {{math|1=''β'' = 4/3}}, {{math|1=''γ'' = 1 = ''δ''}}. Assume {{mvar|x}}, {{mvar|y}} quantify thousands each. Circles represent prey and predator initial conditions from {{mvar|x}} = {{mvar|y}} = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2).

==Dynamics of the system==
In the model system, the predators thrive when prey is plentiful but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a [[population cycle]] of growth and decline.

===Population equilibrium===
Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0:
<math display="block">x(\alpha - \beta y) = 0,</math>
<math display="block">-y(\gamma - \delta x) = 0.</math>

The above system of equations yields two solutions:
<math display="block">\{y = 0,\ \  x = 0\}</math>
and
<math display="block">\left\{y = \frac{\alpha}{\beta},\ \  x = \frac{\gamma}{\delta} \right\}.</math>

Hence, there are two equilibria.

The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters ''α'', ''β'', ''γ'', and ''δ''.

===Stability of the fixed points===
The stability of the fixed point at the origin can be determined by performing a [[linearization]] using [[partial derivative]]s.

The [[Jacobian matrix]] of the predator–prey model is
<math display="block">J(x, y) = \begin{bmatrix} 
 \alpha - \beta y & -\beta x \\
 \delta y & \delta x - \gamma
\end{bmatrix}.</math>
and is known as the [[community matrix]].

====First fixed point (extinction)====
When evaluated at the steady state of {{nowrap|(0, 0)}}, the Jacobian matrix {{mvar|J}} becomes
<math display="block">J(0, 0) = \begin{bmatrix}
 \alpha & 0 \\
 0 & -\gamma
\end{bmatrix}.</math>

The [[eigenvalue]]s of this matrix are
<math display="block">\lambda_1 = \alpha, \quad \lambda_2 = -\gamma.</math>

In the model {{mvar|α}} and {{mvar|γ}} are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a [[saddle point]].

The instability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

====Second fixed point (oscillations)====
Evaluating ''J'' at the second fixed point leads to
<math display="block">J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix}
 0 & -\frac{\beta \gamma}{\delta} \\
 \frac{\alpha \delta}{\beta} & 0
\end{bmatrix}.</math>

The eigenvalues of this matrix are
<math display="block">\lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}.</math>

As the eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y)</math> on orbits. Thus orbits about the fixed point are closed and [[Dynamical system#Conjugation results|elliptic]], so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency <math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> and period <math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>.

As illustrated in the circulating oscillations in the figure above, the level curves are closed [[orbit (dynamics)|orbit]]s surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without [[Damping ratio|damping]] around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>.

The value of the [[constant of motion]] {{math|''V''}}, or, equivalently, {{math|1=''K'' = exp(−''V'')}}, <math>K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x}</math>, can be found for the closed orbits near the fixed point.

Increasing {{math|''K''}} moves a closed orbit  closer to the fixed point. The largest value of the constant {{math|''K''}} is obtained by solving the optimization problem
<math display="block">y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max_{x,y>0}.</math>
The maximal value of ''K'' is thus attained at the stationary (fixed) point <math>\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)</math> and amounts to 
<math display="block">K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math>
where {{math|''e''}} is [[e (mathematical constant)|Euler's number]].

==See also==
{{div col|colwidth=20em}}
*[[Competitive Lotka–Volterra equations]]
*[[Generalized Lotka–Volterra equation]]
*[[Mutualism and the Lotka–Volterra equation]]
*[[Community matrix]]
*[[Population dynamics]]
*[[Population dynamics of fisheries]]
*[[Nicholson&ndash;Bailey model]]
*[[Reaction–diffusion system]]
*[[Paradox of enrichment]]
*[[Lanchester's laws]], a similar system of differential equations for military forces
*[[Random generalized Lotka–Volterra model]]
*[[Consumer-resource model]]
{{div col end}}

==Notes==
{{Reflist|30em}}

==Further reading==
*{{cite book |last1=Hofbauer |first1=Josef |last2=Sigmund |first2=Karl |author-link2=Karl Sigmund |chapter=Dynamical Systems and Lotka–Volterra Equations |title=Evolutionary Games and Population Dynamics |location=New York |publisher=Cambridge University Press |year=1998 |isbn=0-521-62570-X |pages=1–54 }}
*{{cite book|title=Understanding Nonlinear Dynamics|url=https://archive.org/details/understandingnon0000kapl|url-access=registration|first1=Daniel|last1=Kaplan|first2=Leon|last2=Glass|author-link2=Leon Glass|location=New York|publisher=Springer|year=1995|isbn=978-0-387-94440-1}}
*{{cite book|first=E. R.|last=Leigh|year=1968|chapter=The ecological role of Volterra's equations|title=Some Mathematical Problems in Biology}} &ndash; a modern discussion using [[Hudson's Bay Company]] data on [[lynx]] and [[hare]]s in [[Canada]] from 1847 to 1903.
*{{cite book|first=J. D.|last=Murray|title=Mathematical Biology I: An Introduction|location=New York|publisher=Springer|year=2003|isbn=978-0-387-95223-9}}'
*Stefano Allesina's Community Ecology course lecture notes: https://stefanoallesina.github.io/Theoretical_Community_Ecology/

==External links==
{{Commons category}}
*From the ''[[Wolfram Demonstrations Project]]'' — requires [http://demonstrations.wolfram.com/download-cdf-player.html CDF player (free)]:
**[http://demonstrations.wolfram.com/PredatorPreyEquations Predator–Prey Equations]
**[http://demonstrations.wolfram.com/PredatorPreyModel Predator–Prey Model]
**[http://demonstrations.wolfram.com/PredatorPreyDynamicsWithTypeTwoFunctionalResponse Predator–Prey Dynamics with Type-Two Functional Response]
**[http://demonstrations.wolfram.com/PredatorPreyEcosystemARealTimeAgentBasedSimulation Predator–Prey Ecosystem: A Real-Time Agent-Based Simulation]
* [https://espadrine.github.io/lotka-volterra/ Lotka-Volterra Algorithmic Simulation] (Web simulation).

{{modelling ecosystems|expanded=other}}
{{Authority control}}

{{DEFAULTSORT:Lotka-Volterra Equation}}
[[Category:Eponymous equations of mathematics]]
[[Category:Predation]]
[[Category:Ordinary differential equations]]
[[Category:Fixed points (mathematics)]]
[[Category:Population models]]
[[Category:Mathematical modeling]]
[[Category:Community ecology]]
[[Category:Population ecology]]