# Lotka-Volterra model

**Dynamic equations**

Prey: \(\eqalign{\frac{\mathrm{d} X}{\mathrm{d} t} =}\) yr^{-1} \(X - \) yr^{-1} \(XY\)\(-\) yr^{-1} \(X^2\)

Predators: \(\eqalign{\frac{\mathrm{d} Y}{\mathrm{d} t} =}-\) yr^{-1}\(Y +\)\(\times\) yr^{-1}\(XY\)

**Initial state**
\( \qquad\) Prey: \(X^0=\) animals\(, \qquad\) Predators: \(Y^0=\) animals

**Steady state**
\( \qquad X_2^\infty =\) animals \(, \qquad Y_2^\infty =\) animals\(\qquad\) ()

**Population development**

**Phase space**

### The formal equations

The following system of differential equations is the *Lotka-Volterra model* describing the interactions between two groups of animals: predators and prey.

\(X\) is the number of prey animals, \(Y\) the number of predators. \(k_1\) is the prey growth rate. \(k_2\) is the predator death rate. \(k_3\) is the *predation rate*, i.e. it describes the interaction between predator and prey animals.

By adjusting \(\alpha\), we can describe how many predators are “created” from each eaten prey animal. If \(\alpha=1\), each eaten prey animal results in one new predator being added. It is more realistic, however, to set \(\alpha\) to a value smaller than one, i.e. to assume that it needs more than one eaten prey animal to add another predator.

The term marked red is an addition to the basic model that adds self-interaction between prey animals, i.e. it causes the prey population to decrease if it gets too large (for instance, due to increased competition for scarce food).