Editing Lotka–Volterra equations (section)

{{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]].