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| -rw-r--r-- | examples/lotka_volterra/model.modelica | 5 | ||||
| -rw-r--r-- | examples/lotka_volterra/page.md | 31 |
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diff --git a/examples/lotka_volterra/model.modelica b/examples/lotka_volterra/model.modelica index 1ab761e..706ebc3 100644 --- a/examples/lotka_volterra/model.modelica +++ b/examples/lotka_volterra/model.modelica @@ -1,10 +1,11 @@ model LotkaVolterra + "Predator-Prey equations describing population dynamics of two biological species" parameter Real alpha; parameter Real beta; parameter Real delta; parameter Real gamma; - Real x; - Real y; + Real x "population of prey"; + Real y "population of predator"; equation der(x) = (alpha * x) - (beta * x * y); der(y) = (delta * x * y) - (gamma * y); diff --git a/examples/lotka_volterra/page.md b/examples/lotka_volterra/page.md index 8b85919..1ae891d 100644 --- a/examples/lotka_volterra/page.md +++ b/examples/lotka_volterra/page.md @@ -1,8 +1,35 @@ -This is a wikipage! +The Lotka–Volterra equations, also known as the predator–prey equations, are a +pair of first-order, [non-linear](https://en.wikipedia.org/wiki/Non-linear), +differential equations frequently used to describe the dynamics of biological +systems in which two species interact, one as a predator and the other as prey. + +The Lotka–Volterra system of equations is an example of a Kolmogorov +model, which is a more general framework that can model the dynamics of +ecological systems with predator-prey interactions, competition, disease, and +mutualism. + +## Solutions to the equations + +The equations have periodic solutions and do not have a simple expression in +terms of the usual trigonometric functions, although they are quite +tractable. + +If none of the non-negative parameters α,β,γ,δ vanishes, three can be absorbed +into the normalization of variables to leave but merely one behind: Since the +first equation is homogeneous in x, and the second one in y, the parameters β/α +and δ/γ, are absorbable in the normalizations of y and x, respectively, and γ +into the normalization of t, so that only 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 with the population of predators trailing that of prey by 90° in the +cycle. ## References +Body text taken from Wikipedia. + * [Mathworld](http://mathworld.wolfram.com/Lotka-VolterraEquations.html) * [Wikipedia](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations) - +* [Population dynamics of fisheries](https://en.wikipedia.org/wiki/Population_dynamics_of_fisheries) |