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.

$$\begin{eqnarray}\eqalign{\frac{\mathrm{d} X}{\mathrm{d} t} &=&&k_1X &-& k_3 X Y &\color{red}{-}& \color{red}{k_4 X^2} \\ \frac{\mathrm{d} Y}{\mathrm{d} t} &=&-&k_2 Y &+& \alpha k_3 X Y&&}\label{eq:LotkaVolterra}\end{eqnarray}$$

\(X\) 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).