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Mathematical Medicine and Biology Advance Access originally published online on April 28, 2008
Mathematical Medicine and Biology 2008 25(1):1-20; doi:10.1093/imammb/dqm011
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Control of invasive hosts by generalist parasitoids

Christelle Magal{dagger}

Faculté des Sciences, Université de Tours, IRBI UMR CNRS 6035, 37200 Tours, France

Chris Cosner and Shigui Ruan

Department of Mathematics, University of Miami, PO Box 249085, Coral Gables, FL 33124-4250, USA

J. Casas

Faculté des Sciences, Université de Tours, IRBI UMR CNRS 6035, 37200 Tours, France

{dagger} Email: christelle.magal{at}univ-tours.fr

Received on May 7, 2007. Revised on October 8, 2007. Accepted on December 3, 2007.

This article was motivated by the invasion of leaf-mining microlepidopteron attacking horse chestnut trees in Europe and the need for a biological control. Following Owen & Lewis (2001, Bull. Math. Biol., 63, 655–684), we consider predation of leafminers by a generalist parasitoid with a Holling Type II functional response. We first identified six equilibrium points and discussed their stabilities in the non-spatial model. The model always predicts persistence of the parasitoid. Depending on the parameter values, the model may predict that the host persists and goes extinct or there is something like an Allee effect where the outcome depends on the initial host density. Special cases were also studied for small carrying capacities leading to complex dynamical behaviours. Then, numerical simulations of the spatial reaction--diffusion model enabled us to identify the conditions for which the leafminer's advance can be stopped and reversed by parasitoids. Compared to the ordinary differential equation model, the incorporation of space, combined with the polyphagy of the parasitoid, leads to a decrease of the parameter domain of coexistence. This is in stark to several other models in which space promotes coexistence by enabling hosts to escape.

Keywords: ODE; PDE; invasion process; biological control; host–parasitoid system; generalist parasitoid; apparent competition


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