Prey dispersal and predator persistence

Abstract

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 tevet 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.

Effects of Predator and Prey Dispersal on Success or Failure of Biological Control

“…With all prey patches available to the predator, the prey became extinct but coexistence was possible using barriers to reduce the dispersal of the predators and allowing more chances for the prey to disperse. The effects of dispersal on the outcome of predatorprey models has also been investigated by Crowley (1981), Jansen and Sabelis (1992), and Pels and Sabelis (1999), amongst others.…”

Local and distant prey‐related cues influence when an acarine predator leaves a prey patch

“…Although the predator-prey interaction is locally unstable, it may persist on a regional scale due to repeated dispersal and colonization of new patches (Diekmann et al, 1988;Nachman, 1987Nachman, , 1988Nachman, , 1991Sabelis et al, 1991;Walde, 1991Walde, , 1994Jansen & Sabelis, 1992). Models of the mite predator-prey interaction (Sabelis et al, 1998;Pels & Sabelis, 1999) predict different local and regional persistence depending on predator dispersal tendencies (e.g., leaving rates).…”

On the role of body size in a tri-trophic metapopulation model

“…Model (1-5) differs from the nlodels discussed in Sabelis et al (1991), Jansen and Sabelis (1992), a.nd Jansen (1995) for two rea.sons: first, because both species are dispersed at the same time and second, and nlost importa,nt, because the rates of inva.sion of free an prey patches do not increa,se indefinitely with the number of such pa.tc11es. On the contrary, the models discussed in t,he above-mentioned papers are c1la.racterized by rates of invasion proportional to the number of inva,dable patches.…”

“…(1991), Jansen and Sabelis (1992), and Jansen (1995). The aim is t o establish the role played by prey and predator body sizes in determining the asy~nptotic behavior of the system.…”