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)