View source for Lotka–Volterra equations

{{About|the predator-prey equations|the competition equations|Competitive Lotka–Volterra equations}}
{{short description|Pair of equations modelling predator-prey cycles in biology}}
The '''Lotka–Volterra equations''', also known as the '''predator–prey equations''', 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 populations change through time according to the pair of equations:
:<math>
\begin{align}
 \frac{dx}{dt} &= \alpha x - \beta x y, \\
 \frac{dy}{dt} &= \delta x y - \gamma y,
\end{align}
</math>
where
:{{mvar|x}} is the number of prey (for example, [[rabbit]]s);
:{{mvar|y}} is the number of some [[Predation|predator]] (for example, [[fox]]es);
:<math>\tfrac{dy}{dt}</math> and <math>\tfrac{dx}{dt}</math> represent the instantaneous growth rates of the two populations;
:{{mvar|t}} represents time;
:{{math|''α'', ''β'', ''γ'', ''δ''}} are positive real [[parameter]]s describing the interaction of the two [[species]].

The Lotka–Volterra system of equations is an example of a [[Kolmogorov model]],<ref>{{cite book |last=Freedman |first=H. I. |title=Deterministic Mathematical Models in Population Ecology |publisher=[[Marcel Dekker]] |year=1980}}</ref><ref>{{cite book |last=Brauer |first=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 }}</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]].

==History==
[[File:Milliers fourrures vendues en environ 90 ans odum 1953 en.jpg|left|upright=2.5|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.]]
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}}</ref><ref name="Goelmany">{{cite book|last=Goel|first=N. S.|display-authors=etal|title=On the Volterra and Other Non-Linear Models of Interacting Populations|location=|publisher=Academic Press|year=1971}}</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|df=|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/?id=8GsEAAAAYAAJ#v=onepage&q=&f=false|title=Notice sur la loi que la population poursuit dans son accroissement|journal=Corresp. Mathématique et Physique|volume=10|issue=|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}}</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|location=|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|issue=|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.|location=|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. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.<ref>{{cite book|last=Kingsland|first=S.|title=Modeling Nature: Episodes in the History of Population Ecology|location=|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–McArthur model.<ref>{{cite journal|last=Rosenzweig|first=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}}</ref> Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, 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}}</ref> and the moose and wolf populations in [[Isle Royale National Park]].<ref>{{cite journal|last=Jost|first=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>

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|last=Arditi|first=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}}</ref> The validity of prey- or ratio-dependent models has been much debated.<ref>{{cite journal|last=Abrams|first=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}}</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.|authorlink=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}}</ref> or 1967.<ref>{{cite book|last=Goodwin|first=R. M.|chapter=A Growth Cycle|title=Socialism, Capitalism and Economic Growth|url=https://archive.org/details/socialismcapital0000fein|url-access=registration|editor-last=Feinstein|editor-first=C. H.|publisher=[[Cambridge University Press]]|year=1967}}</ref><ref>{{cite journal|last=Desai|first=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}}</ref>

==Physical meaning of the equations==
The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:<ref>{{Cite web|url=http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html|title=PREDATOR-PREY DYNAMICS|website=www.tiem.utk.edu|access-date=2018-01-09}}</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.

As differential equations are used, the solution 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|last=Cooke|first=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>
{{see also|Competitive Lotka–Volterra equations}}

===Prey===
When multiplied out, the prey equation becomes
:<math>\frac{dx}{dt} = \alpha x - \beta x y.</math>

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 ''αx''. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by ''βxy''. If either {{mvar|x}} or {{mvar|y}} is zero, then there can be no predation.

With these two terms the 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.

===Predators===
The predator equation becomes
:<math>\frac{dy}{dt} = \delta xy - \gamma y.</math>

In this equation, ''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). ''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.

==Solutions to the equations==
The equations have [[periodic function|periodic]] solutions and do not have a simple expression in terms of the usual [[trigonometric function]]s, although they are quite tractable.<ref>{{cite journal|last=Steiner|first=Antonio|last2=Gander|first2=Martin Jakob|year=1999|title=Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich|journal=Il Volterriano|volume=7|issue=|pages=32–44|url=http://archive-ouverte.unige.ch/unige:6300/ATTACHMENT01}}</ref><ref>{{cite journal|last=Evans|first=C. M.|last2=Findley|first2=G. L.|title=A new transformation for the Lotka-Volterra problem|journal=Journal of Mathematical Chemistry|volume=25|issue=|pages=105–110|year=1999|doi=10.1023/A:1019172114300}}</ref>

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

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|location=|publisher=Springer–Verlag|year=1983}}</ref> with the population of predators trailing that of prey by 90° in the cycle.
[[Image:Lotka Volterra dynamics.svg|center]]
{{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 baboons and cheetahs 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 baboon (prey) and a cheetah (predator). If the initial conditions are 10 baboons and 10 cheetahs, one can plot the progression of the two species over time; given the parameters that the growth and death rates of baboon are 1.1 and 0.4 while that of cheetahs 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 number 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'' and ''y''. 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 problem with this ''as a biological model'': For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs 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}}</ref>

===Phase-space plot of a further example===
[[Image:Lotka-Volterra orbits 01.svg|alt=|left|thumb|300x300px]]A less extreme example covers: 
{{mvar|α}} = 2/3, {{mvar|β}} = 4/3, {{mvar|γ}} = 1 = {{mvar|δ}}. Assume {{math|''x'', ''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 there are plentiful prey 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 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>x(\alpha - \beta y) = 0,</math>
:<math>-y(\gamma - \delta x) = 0.</math>

The above system of equations yields two solutions:

:<math>\{y = 0,\ \  x = 0\}</math>

and

:<math>\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>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 (0,&nbsp;0), the Jacobian matrix ''J'' becomes

:<math>J(0, 0) = \begin{bmatrix}
 \alpha & 0 \\
 0 & -\gamma
\end{bmatrix}.</math>

The [[eigenvalue]]s of this matrix are

:<math>\lambda_1 = \alpha, \quad \lambda_2 = -\gamma.</math>

In the model ''α'' and ''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 stability 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>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>\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 others, this fixed point is [[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]] around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>.

The value of the [[constant of motion]] ''V'', or, equivalently, ''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 ''K'' moves a closed orbit  closer to the fixed point. The largest value of the constant ''K'' is obtained by solving the optimization problem
:<math>y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max\limits_{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>K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math>
where ''e'' is [[e (mathematical constant)|Euler's number]].

==See also==
*[[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

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

==References==
*{{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|title=Understanding Nonlinear Dynamics|url=https://archive.org/details/understandingnon0000kapl|url-access=registration|first=Daniel|last=Kaplan|first2=Leon|last2=Glass|authorlink2=Leon Glass|location=New York|publisher=Springer|year=1995|isbn=978-0-387-94440-1}}
*{{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}}
*{{cite journal|first=James A.|last=Yorke|first2=William N. Jr.|last2=Anderson|title=Predator-Prey Patterns (Volterra-Lotka equations)|journal=[[PNAS]]|volume=70|issue=7|pages=2069–2071|year=1973|jstor=62597|doi=10.1073/pnas.70.7.2069|url=http://www.pnas.org/content/70/7/2069.full.pdf}}
*{{cite journal|first=J.|last=Llibre|first2=C.|last2=Valls|title=Global analytic first integrals for the real planar Lotka-Volterra system|journal=J. Math. Phys.|volume=48|issue=3|pages=033507|year=2007|doi=10.1063/1.2713076}}

==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}}

{{DEFAULTSORT:Lotka-Volterra Equation}}
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