= Solution
This is the <Nicholson-Bailey model> for a <host>-<parasitoid> interaction with discrete generations. The factor $r$ is the host multiplication factor in the absence of parasitism; $a$ measures effective attack or searching intensity per parasitoid; and $b$ is the number of next-generation parasitoids produced per attacked host, with any survival or unit conversion absorbed into it.
If the number of attacks on a host has a <Poisson distribution> with <expected value> $aP_t$, then the <probability> of no attack is $e^{-aP_t}$. Consequently $e^{-aP_t}$ is the surviving, unparasitized host fraction, and $1-e^{-aP_t}$ is the attacked fraction. The two update equations count host descendants from surviving hosts and parasitoid descendants from attacked hosts, respectively.
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