seeds to safe sites (H. Godfnez-Alvarez, A. Rojas-Marti'nez, and A. Valiente-Banuet, unpublished observations).
The maintenance of cactus populations under natural conditions is affected by various interactions with other organisms as well as by abiotic factors, such as rainfall and solar radiation (Fig. 6.1). The assessment of the relative importance of each factor provides information on the main causes that affect the population dynamics of these plants. In this respect, matrix models are a useful tool to quantify the relative contribution made by different life stages to the rate of population increase. In such demographic models, individuals are grouped into size categories, and a projection matrix is constructed according to the survival, growth, and reproduction probabilities of the individuals in each category. This projection matrix is formed by three parts: (1) the first row includes the fecundity values for all of the reproductive classes, (2) the main diagonal includes the probabilities that individuals remain in the same size class after one year, and (3) the first lower subdiagonal incorporates the probabilities that individuals grow to the next size class after one year. The solution of this model permits one to obtain the rate of population increase (the largest positive eigenvalue), the stable size distribution
(right eigenvector), and the reproductive values (left eigenvector; De Kroon et al. 1986). Moreover, elasticity can be analyzed for the critical stages of the life cycle. Elasticity is a relative measure, between 0 and 1 (or 0 and 100%), of the change in the rate of population increase in response to changes in matrix values (De Kroon et al. 1986).
Studies with this approach have been conducted with the giant columnar cacti C. gigantea (Silvertown et al. 1993), Neobuxbaumia tetetzo (Godínez-Alvarez et al. 1999), and P pringlei. These studies show similar demographic patterns in which the survivorship of young individuals makes a higher contribution to the rate of population increase than does growth and reproduction of older plants (Fig. 6.1). The elasticity values of survivorship varies among species, from 86% for Ppringlei, 91% for N. tetetzo, to 99% for C. gigantea (Silvertown et al. 1993; Godínez-Alvarez et al. 1999).
Elasticity analyses of N. tetetzo and P pringlei indicate that survivorship of immature and mature individuals have the highest values, ranging from 20 to 60% (Fig. 6.2). Survivorship of these categories is affected by competitive interactions with other plants of the same and/or different species (McAuliffe 1984b; McAuliffe and Janzen 1986; Valiente-Banuet et al. 1991b; Flores-Martínez et al. 1998). On the other hand, elasticity values for the rest of the cat-
Flower and bud predation Pollination Herbivores
Nurse plants Seed production Seed dispersal Soil moisture
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