Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 13B / a

$T$ is the interval between generations or censuses, and $r$ the prey multiplication without predators. The factor $e^{-aP}$ can be the probability of escaping a Poisson number of encounters with mean $aP$, so $a$ measures attack intensity. The factor $s$ converts prey resources into predator recruitment, and $b$ modifies the fraction of prey contributing no recruitment. At $b=1$ predators recruit from attacked prey; $b<1$ adds background recruitment, even at $P=0$. A population interpretation valid at every nonnegative state needs $b\le 1$, since otherwise the second equation gives negative recruitment near $P=0$.