# examples/LotkaVolterra

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Predator-Prey equations describing population dynamics of two biological species

$$\begin{array}{rcl} \frac{ d x }{ dt } & = & \left(\alpha \cdot x - \beta \cdot x \cdot y\right) \\ \frac{ d y }{ dt } & = & \left(\delta \cdot x \cdot y - \gamma \cdot y\right) \\ \end{array}$$
Variable Type Description Value (default) Datatype
$$\alpha$$ parameter  Real
$$\beta$$ parameter  Real
$$\delta$$ parameter  Real
$$\gamma$$ parameter  Real
$$x$$ 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.

## References

Body text taken from Wikipedia.

## Modelica Code

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;


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