Prey dispersal and predator persistence

To understand how patchiness influences population dynamics of a tri-trophic interaction, a tractable model is formulated in terms of differential equations. Motivated by the structure of systems such as plants, phytophagous mites and predatory mites, the model takes dispersal into account at the middle trophic level. The effect of dispersal for the middle level in a tri-trophic system could be either stabilising or destabilising since the middle level acts both as prey and as predator. First a simple model with logistic growth for the lowest level is formulated. A model with logistic growth for the lowest level and instantaneous dispersal has a globally stable three-species equilibrium, if this equilibrium exists. Addition of a middle level dispersal phase of non-negligible duration influences equilibrium stability. In the absence of the top trophic level a limit cycle can occur, caused by the delay that exists in the reaction of the middle level to the changes in the lowest level. With low predator efficiency, it is also possible to have an unstable three-species equilibrium. So addition of a middle level dispersal phase of non-negligible duration can work destabilising. Next the persistence of the third trophic level is studied. Even when the three-species equilibrium exists, the third trophic level need not be persistent. A two-species limit cycle can keep its stability when a three-species equilibrium exists; the system is then bistable. It is argued that such a bistability can offer an alternative explanation for pesticide-induced outbreaks of spider mites and failure of predator introduction.