Predator-Prey Model

Question

Can anyone explain the biological interpretation on the right hand side of these equations please?

$$ \begin{array}{r c l} \frac{dN}{dt} & = & rN\left(1-\frac{N}{K}\right)-\alpha \frac{NP}{\beta P + \gamma N} \\ \frac{dP}{dt} & = & \epsilon \frac{NP}{\beta P + \gamma N} - \delta P \end{array}$$

with $N(t)$ the number of prey and $P(t)$ the number of predators.

Answer 1

Each equation has two main terms on the right side, they can be interpreted as follows:

species	term 1	term 2
$N$	internal $N$ dynamic, typical logistic growth to carrying capacity $K$	death due to predator $P$ encounters, scaled
$P$	births increase with prey catches, scaled as above	death term

The encounter term is scaled to be more realistic than the pure product $NP$. For $P\ll N$ it is mostly proportional to $P$, reflecting the fact that a predator just hunts as much as it can consume. In the exceptional situation that $N$ becomes very small, the encounters become proportional to $N$, which also seems a sensible behavior.