diff options
Diffstat (limited to 'examples/lotka_volterra_predator_prey')
| -rw-r--r-- | examples/lotka_volterra_predator_prey/examples.toml | 11 | ||||
| -rw-r--r-- | examples/lotka_volterra_predator_prey/model.modelica | 12 | ||||
| -rw-r--r-- | examples/lotka_volterra_predator_prey/page.md | 35 |
3 files changed, 58 insertions, 0 deletions
diff --git a/examples/lotka_volterra_predator_prey/examples.toml b/examples/lotka_volterra_predator_prey/examples.toml new file mode 100644 index 0000000..0f5fb85 --- /dev/null +++ b/examples/lotka_volterra_predator_prey/examples.toml @@ -0,0 +1,11 @@ + +[examples] + + [examples.mathworld] + x = 10 + y = 5 + alpha = 1.5 + beta = 1 + gamma = 1 + t = [0, 20] + diff --git a/examples/lotka_volterra_predator_prey/model.modelica b/examples/lotka_volterra_predator_prey/model.modelica new file mode 100644 index 0000000..706ebc3 --- /dev/null +++ b/examples/lotka_volterra_predator_prey/model.modelica @@ -0,0 +1,12 @@ +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_predator_prey/page.md b/examples/lotka_volterra_predator_prey/page.md new file mode 100644 index 0000000..1ae891d --- /dev/null +++ b/examples/lotka_volterra_predator_prey/page.md @@ -0,0 +1,35 @@ + +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) |