Editing Lotka–Volterra equations (section)

==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).