- Lottery coexistence models extended to plants with disjoint generations - 161 space becomes identical to character space, and its util- ity is lost. To retain explanatory power for the niche concept, we may have to abandon the requirement of a separate niche for each member of a set of coexisting species. We prefer to think as follows: the niche concept satisfactorily explains the coexistence of species that demonstrably differ in the way they make use of re- sources. However, ecologically similar coexisting spe- cies occupy only one niche; this is explained in a differ- ent way, for example through stochastic models of re- cruitment for mechanisms whereby organisms may co- exist in a single niche (as opposed to deterministic models of competition which yield mechanisms whereby at equilibrium any niche can only have a sole occupant). The idea that plant communities often consist of many more plant species than niches, i.e. that in such commu- nities most of the species share their niche with several others, is becoming common in the literature (Silver- town & Law 1987; van der Maarel & Sykes 1993; but see Chesson 1991 for a dissenting view). We propose that fynbos and kwongan should be added to the list of communities where a large number of species coexist by sharing a small number of niches. In verbal form, stochastic models of recruitment have existed for many years (e.g. Andrewartha & Birch 1954; Sale 1977). Fagerström & Ågren (1979, 1980), who were interested in phenological spread as a mecha- nism for coexistence, formulated a simple mathematical model for plants where the environment consists of a set of sites that can be occupied by at most one adult of any species. Adults die at species-specific rates, and seed- lings take their place on a random basis, with probabili- ties determined by the relative frequency of seedlings present for each species as well as by competitive abil- ity. Fluctuations in recruitment success among species are determined entirely by the seedling stage: a species succeeds according to the number of seeds present at a site together with their relative ability to exploit the environmental conditions they find at the time an adult dies. Fagerström & Ågren (1979) found that, when generations do not overlap, the superior competitor will Lottery coexistence models extended to plants with disjoint generations Laurie, Henri 1* & Cowling, Richard M. 2 1 Department of Applied Mathematics, University of Cape Town, Private Bag 7700, Rondebosch, South-Africa; 2 Department of Botany, University of Cape Town, Private Bag 7700, Rondebosch, South-Africa; * Tel. +21 650 2332; Fax +21 650 2334; E-mail HENRI@MATHS.UCT.AC.ZA Abstract. Neither conventional niche theory nor current lot- tery models offer a satisfactory theoretical scope for model- ling coexistence of species with disjoint generations. South- African fynbos and Australian kwongan include many species which are killed by, and recruit only after, fire. We propose a density-dependent lottery model which accommodates the unusual demographics of these species. We show that coexist- ence requires density dependence in recruitment. The result applies to a wider class of populations than the one considered here. It is applied to non-resprouting species in fynbos and kwongan. We show that the lottery assumption of recruitment in proportion to propagules is often satisfied, while the pro- duction of such propagules is often density-dependent, and we discuss some evidence of mechanisms whereby this may occur. Keywords: Demography; Fire; Fynbos; Kwongan; Popula- tion model; Recruitment. Introduction Niche differentiation cannot always account for spe- cies richness in plant communities. In highly species- rich communities, coexisting plants are often highly similar in growth form, overall trophic requirements, phenology and pollination mode (Grubb 1977; Shmida & Ellner 1984). Examples include tropical rainforests (Hubbell & Foster 1986) and the fire-prone shrublands of the mediterranean-climate regions of southwestern Africa: fynbos, and southwestern Australia: kwongan (Cowling 1987; Lamont & Bergl 1991; Bond et al. 1992). For these communities, it is very difficult to separate coexisting species along conventional niche axes (Silvertown & Law 1987). Unconventional niche axes have been suggested. For example, Cody (1986) hypothesized that fynbos Protea spp. are differentiated by leaf size and shape; Bond et al. (1992) proposed various regeneration niches (sensu Grubb 1977). Apart from difficulties in testing such hypotheses, the lack of theory that can predict a priori the set of possible niche axes makes such hypotheses seem ad hoc, not to say circular: whatever is different among coexisting plants becomes a candidate for a niche axis. In this way niche Journal of Vegetation Science 5: 161-168, 1994 © IAVS; Opulus Press Uppsala. Printed in Sweden