diff options
Diffstat (limited to 'examples/lotka_volterra')
| -rw-r--r-- | examples/lotka_volterra/examples.toml | 11 | ||||
| -rw-r--r-- | examples/lotka_volterra/model.modelica | 12 | ||||
| -rw-r--r-- | examples/lotka_volterra/page.md | 35 |
3 files changed, 0 insertions, 58 deletions
diff --git a/examples/lotka_volterra/examples.toml b/examples/lotka_volterra/examples.toml deleted file mode 100644 index 0f5fb85..0000000 --- a/examples/lotka_volterra/examples.toml +++ /dev/null @@ -1,11 +0,0 @@ - -[examples] - - [examples.mathworld] - x = 10 - y = 5 - alpha = 1.5 - beta = 1 - gamma = 1 - t = [0, 20] - diff --git a/examples/lotka_volterra/model.modelica b/examples/lotka_volterra/model.modelica deleted file mode 100644 index 706ebc3..0000000 --- a/examples/lotka_volterra/model.modelica +++ /dev/null @@ -1,12 +0,0 @@ -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 "population of prey"; - Real y "population of predator"; -equation - der(x) = (alpha * x) - (beta * x * y); - der(y) = (delta * x * y) - (gamma * y); -end LotkaVolterra; diff --git a/examples/lotka_volterra/page.md b/examples/lotka_volterra/page.md deleted file mode 100644 index 1ae891d..0000000 --- a/examples/lotka_volterra/page.md +++ /dev/null @@ -1,35 +0,0 @@ - -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) |