A Constant Introduction of Additional Food Does Not Yield Pest Eradication in Finite Time

Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the introduced predators efficacy can be boosted by providing them with an additional food source. The current literature \cite{SP11} claims that if the additional food is of sufficient constant quantity and quality then pest eradication is possible in \emph{finite} time. The purpose of the current manuscript is to show that to the contrary, pest eradication is \emph{not possible} in finite time, for \emph{any} quantity and quality of constant additional food. However for a density dependent quantity of additional food, we show pest eradication in finite time is indeed possible. Our results have large scale implications for the effective design of biological control methods involving additional food.