T(w)o patch or not t(w)o patch: A novel additional food model

A number of top down bio-control models have been proposed where the predators' efficacy is enhanced via the provision of additional food (AF). These can lead to infinite time \emph{blow-up} of the predator . Predator interference, predator competition, refuge, pest defense and cannibalism have been proposed to achieve pest extinction, while keeping the predator population bounded, but entail \emph{quadratic} or higher order damping terms. Thus an AF model with monotone pest dependent functional response, devoid of these mechanisms, cannot yield the pest extinction state. In the current manuscript we propose a novel patch driven bio-control model, with monotone pest dependent functional response. Here a predator is introduced into a ``patch" such as a prairie strip with AF, and then disperses into a neighboring ``patch" such as a crop field, to target a pest. Assuming only \emph{linear} dispersal in the predator, we show, (i) blow-up in the predator can be prevented in both the prairie strip and the crop field, (ii) the pest extinction state is attainable in the crop field, (iii) the AF model with patch structure can keep pest densities lower than the AF model without patch structure and/or the classical top-down bio-control model, (iv) the AF model with patch structure possesses rich dynamics including chaos. In particular, one observes ``patch specific chaos" - the pest occupying the crop-field can oscillate chaotically, while the pest in the prairie strip oscillates periodically. We discuss these results in light of bio-control strategies that would utilize prairie strips.