J. theor. Biol. (2003) 221, 61–77 doi:10.1006/jtbi.2003.3171, available online at http://www.idealibrary.com on Coexistence in a Competitive Parasitoid–host System DerikCastillo n w andJorgeX.Velasco-Herna L ndezz wInstituto de Ecologı´a, Universidad Nacional Auto´noma de Me´xico, Apartado postal 70-275, Ciudad Universitaria, C.P. 04510, Mexico and zDepartamento de Matema´ticas UAM-Iztapalapa and Programa de Matema´ticas Aplicadas y Computacio´n, Instituto Mexicano del Petro´leo, Eje Central La´zaroCa´rdenas 152, Col. San Bartolo Atepehuaca´n D.F. 07730, Me´xico (Received on 21 December 2001, Accepted in revised form on 18 September 2002) The main objective of this work is to determine the conditions for coexistence and competitive exclusion in a discrete model for a community of three species: a stage-structured host and two competing parasitoids sharing the same host developmental stage. Coexistence of the community of the species is found to depend on the host life history parameters in the first place, and on competitive ability and parasitoid efficiency in the second place. In particular, parasitoids equilibrium densities are defined by the size of the refuge. Extinction is expected with low growth rate and with low adult survival. Host life histories are also associated with oscillations in population density, and depending on the combination of host adult survival from one generation to the next and host growth rate, the minimum of fluctuations approaches zero, implying a higher potential risk of extinction because of stochastic factors. Our results suggest that equally reduced survival of parasitoids in hosts parasitized by both species determines extinction of the parasitoid with lower population density, in contrast to the case when both parasitoids benefit with 50% of all doubly parasitized hosts, leading to the hypothesis that a community where competitors in multiparasitized hosts die, easily becomes extinct. Competitive exclusion is expected for highly asymmetric competitive interactions, independent of population densities, allowing us to hypothesize that coexistence of competitors in systems with limited resources and refuges is associated with a clearly defined competitive hierarchy. r 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction In this work, we study the conditions for coexistence in a community of two competing parasitoids sharing a host in a particular developmental stage. We assume two parasitoid types differing only in searching efficiency and competitive ability defined as the probability that one parasitoid survives competition in a doubly parasitized host. We centered our study first on the role of host life history parameters and second on the role of competitive ability. Competition between parasitoids occurs when two or more parasitoid species share a host (Godfray, 1994; Quicke, 1997). Depending on the specific biological assumptions, this n Corresponding author. Programa de Matema´ticas Aplicadas y Computacio´ n, Instituto Mexicano del Petro´ leo, Eje Central La´zaro Ca´rdenas 152, Col. San Bartolo Atepehuaca´n D.F. 07730, Me´xico. Fax: +52-58-04-46-60. E-mail addresses: derik@miranda.ecologia.unam.mx (D. Castillo), velascoj@imp.mx (J. X. Velasco-Herna´ndez). 0022-5193/03/$35.00 r 2002 Elsevier Science Ltd. All rights reserved.