examples/LotkaVolterramarkdown - original modelica
Predator-Prey equations describing population dynamics of two biological species
|\(x\)||free||population of prey||\(\)||Real|
|\(y\)||free||population of predator||\(\)||Real|
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, 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.
Body text taken from Wikipedia.
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;